We prove that any zero torsion linear map on a nonsolvable real Lie algebra 𝔀0 is an extension of some CR-structure. We then study the cases of 𝔰𝔩(2, ℝ) and the 3-dimensional Heisenberg Lie algebra 𝔫. In both cases, we compute up to equivalence all zero torsion linear maps on 𝔀0, and deduce an explicit description of the equivalence classes of integrable complex structures on 𝔀0×𝔀0.

1. Introduction

Given a real Lie algebra 𝔀0, the determination up to equivalence of zero torsion linear maps from 𝔀0 to 𝔀0 plays an important role in the computation of complex structures on direct products involving 𝔀0 [1]. The direct computation of those maps can be difficult for semisimple 𝔀0, so there is a point in exploring alternative ways, particularly their relation to 𝐢𝑅-structures. For compact 𝔀0, maximal rank 𝐢𝑅-structures have been classified up to equivalence in [2]. In the case of 𝔰𝔲(2), all zero torsion linear maps are extensions of certain 𝐢𝑅-structures (see [1]). One can then ask the natural question whether or not any zero torsion linear map on a nonabelian 𝔀0 is necessarily an extension of some 𝐢𝑅-structure. In the present note, we answer the question in the positive for nonsolvable Lie algebras. Then we make a detailed study of two basic examples: 𝔀0=𝔰𝔩(2,ℝ) in the positive case, and 𝔀0=𝔫 the 3-dimensional Heisenberg Lie algebra in the negative. In both cases, we compute (up to equivalence) all zero torsion linear maps, and the result is used to exhibit a complete set of representatives of equivalence classes of complex structures on 𝔀0×𝔀0.

An interesting direction for future research could be to investigate zero torsion linear maps and 𝐢𝑅-structures on various constructions of Lie algebras, for example like those considered in [3] (see also [4]).

2. Zero Torsion Linear Maps and Extension of 𝐢𝑅-Structures

A 𝐢𝑅-structure on a smooth real manifold 𝑀 is a subbundle 𝒱 of the complexified tangent bundle ℂ𝑇(𝑀) of 𝑀 such that [𝒱,𝒱]βŠ‚π’± (i.e., the space of smooth sections of 𝒱 is closed under commutators) and π’±βˆ©π’±={0} ( βˆ’ denoting here conjugation in ℂ𝑇(𝑀)). The rank or 𝐢𝑅-dimension is the complex dimension of 𝒱. For general background on 𝐢𝑅-structures we refer the reader to [5].

Throughout this section, 𝔀0 will denote any finite-dimensional real Lie algebra, 𝔀 its complexification, and 𝜎 or simply βˆ’ the conjugation in 𝔀 with respect to 𝔀0.

If 𝐺0 is a connected finite dimensional real Lie group, with Lie algebra 𝔀0, left invariant 𝐢𝑅-structures on 𝐺0 are identified to 𝐢𝑅-structures on 𝔀0 in the following sense [6, 7].

Definition 2.1. A rank π‘Ÿ(1β©½π‘Ÿβ©½[(dim𝔀0)/2])𝐢𝑅-structure on 𝔀0 is a π‘Ÿ-dimensional complex subalgebra π”ͺ of 𝔀 such that π”ͺ∩π”ͺ={0}.

If dim𝔀0 is even, a 𝐢𝑅-structure of maximal rank is an (integrable) complex structure.

Now one has the following straightforward lemma.

Lemma 2.2. Let 𝔭 be vector subspace of 𝔀0 and π½π”­βˆΆπ”­β†’π”­ a linear map. Consider the real vector subspace π”ͺ={𝑋;π‘‹βˆˆπ”­} of 𝔀, where 𝑋=π‘‹βˆ’π‘–π½π”­π‘‹ for π‘‹βˆˆπ”­. Denote π”ͺξ…ž=𝑖π”ͺ={𝐽𝔭𝑋+𝑖𝑋;π‘‹βˆˆπ”­}. Then (i)π”ͺ is a complex vector subspace of 𝔀 if and only if 𝐽2𝔭=βˆ’1𝔭;(ii)π”ͺ is stable with respect to the bracket if and only if for any𝑋,π‘Œβˆˆπ”­[]βˆ’ξ€Ίπ½π‘‹,π‘Œπ”­π‘‹,π½π”­π‘Œξ€»π½βˆˆπ”­,𝔭[]βˆ’ξ€Ίπ½π‘‹,π‘Œπ”­π‘‹,π½π”­π‘Œ=𝐽𝔭+𝑋,π‘Œπ‘‹,π½π”­π‘Œξ€».(2.1) In that case, [𝑍𝑋,π‘Œ]= with 𝑍=[𝑋,π‘Œ]βˆ’[𝐽𝔭𝑋,π½π”­π‘Œ].(iii)π”ͺξ…ž is stable with respect to the bracket if and only if for any 𝑋,π‘Œβˆˆπ”­ξ€Ίπ½π”­ξ€»+𝑋,π‘Œπ‘‹,π½π”­π‘Œξ€»π½βˆˆπ”­,𝔭𝐽𝔭+𝑋,π‘Œπ‘‹,π½π”­π‘Œ=𝐽𝔭𝑋,π½π”­π‘Œξ€»βˆ’[].𝑋,π‘Œ(2.2) In that case, [𝑍𝑋,π‘Œ]=βˆ’π‘– with 𝑍=[𝐽𝔭𝑋,π‘Œ]+[𝑋,π½π”­π‘Œ] for 𝑋,π‘Œβˆˆπ”­.

Remark 2.3. When (i) is satisfied, (ii) and (iii) are trivially equivalent since then π”ͺ=π”ͺξ…ž.

Lemma 2.4. A rank π‘ŸπΆπ‘…-structure on 𝔀0 can be defined in an alternative way as (𝔭,𝐽𝔭) where 𝔭 is a 2π‘Ÿ-dimensional (1β©½π‘Ÿβ©½[(dim𝔀0)/2]) vector subspace of 𝔀0 and π½π”­βˆΆπ”­β†’π”­ is a linear map satisfying the 3 conditions 𝐽2𝔭=βˆ’1𝔭,𝐽(2.3)𝔭+𝑋,π‘Œπ‘‹,π½π”­π‘Œξ€»ξ€Ίπ½βˆˆπ”­βˆ€π‘‹,π‘Œβˆˆπ”­,(2.4)𝔭𝑋,π½π”­π‘Œξ€»βˆ’[]𝑋,π‘Œβˆ’π½π”­π½ξ€·ξ€Ίπ”­ξ€»+𝑋,π‘Œπ‘‹,π½π”­π‘Œξ€»ξ€Έ=0βˆ€π‘‹,π‘Œβˆˆπ”­.(2.5)

Proof. Let π”ͺ be a rank π‘ŸπΆπ‘…-structure on 𝔀0. Note first that the taking of the real part is a bijective linear map of the real algebra π”ͺ onto its image 𝔭=β„œπ”ͺ,  dim𝔭=2π‘Ÿ, and there exists a unique linear map π½π”­βˆΆπ”­β†’π”€0 such that π”ͺ={𝑋=π‘‹βˆ’π‘–π½π”­π‘‹;π‘‹βˆˆπ”­}. Now, for π‘‹βˆˆπ”­,  𝑖𝑋=𝐽𝔭𝑋+π‘–π‘‹βˆˆπ‘–π”ͺ=π”ͺ hence π½π”­π‘‹βˆˆπ”­, so that π½π”­βˆΆπ”­β†’π”­. Then (2.3), (2.4), (2.5) follow from Lemma 2.2 and Remark 2.3.
The converse comes again from Lemma 2.2 and Remark 2.3.

Remark 2.5. The condition (2.3) implies det𝐽𝔭=1 and Trace (𝐽𝔭)=0, hence if π‘Ÿ=1, (2.4) and (2.5) follow from (2.3) and can be omitted.

Definition 2.6. A linear map π½βˆΆπ”€0→𝔀0 is said to have zero torsion if it satisfies the condition []βˆ’[][][]𝐽𝑋,π½π‘Œπ‘‹,π‘Œβˆ’π½π½π‘‹,π‘Œβˆ’π½π‘‹,π½π‘Œ=0βˆ€π‘‹,π‘Œβˆˆπ”€0.(2.6)

If 𝐽 has zero torsion and satisfies in addition 𝐽2=βˆ’1,  𝐽 is an (integrable) complex structure on 𝔀0. That means that 𝐺0 can be given the structure of a complex manifold with the same underlying real structure and such that the canonical complex structure on 𝐺0 is the left invariant almost complex structure 𝐽 associated to 𝐽 (for more details, see [8]).

When computing the matrices of the zero torsion maps in some fixed basis (π‘₯𝑗)1⩽𝑗⩽𝑛 of 𝔀0, we will denote by π‘–π‘—βˆ£π‘˜ (1⩽𝑖,𝑗,π‘˜β©½π‘›) the torsion equation obtained by projecting on π‘₯π‘˜ the equation (2.6) with 𝑋=π‘₯𝑖, π‘Œ=π‘₯𝑗.

The automorphism group Aut  𝔀0 of 𝔀0 acts on the set of all zero torsion linear maps and on the set of all complex structures on 𝔀0 by π½β†¦Ξ¦βˆ˜π½βˆ˜Ξ¦βˆ’1 for all Φ∈Aut  𝔀0. It acts also on the set of 𝐢𝑅-structures by (𝔭,𝐽𝔭)↦(Φ𝔭,Ξ¦βˆ˜π½π”­βˆ˜Ξ¦βˆ’1). Two 𝐽,π½ξ…ž (resp., (𝔭,𝐽𝔭), (π”­ξ…ž,𝐽𝔭′)) are said to be equivalent (notation: π½β‰‘π½ξ…ž (resp., 𝐽𝔭≑𝐽𝔭′)) if they are on the same Aut  𝔀0 orbit. This means that the corresponding left invariant 𝐢𝑅-structures on the connected simply connected real Lie group associated to 𝔀0 are intertwined by some Lie group automorphism. It is a stronger notion than 𝐢𝑅-diffeomorphy, where the intertwining is simply required to be a diffeomorphism.

Lemma 2.7. Let π½βˆΆπ”€0→𝔀0 be a linear map, π”ͺ={𝑋=π‘‹βˆ’π‘–π½π‘‹;π‘‹βˆˆπ”€0} and π”ͺξ…ž=𝑖π”ͺ={𝐽𝑋+𝑖𝑋;π‘‹βˆˆπ”€0}; (i)π”ͺ∩π”ͺξ…žξ‚={𝑋;π‘‹βˆˆker(𝐽2+1)}, (ii)π”ͺξ…ž is a real subalgebra of 𝔀 if and only if 𝐽 has zero torsion,(iii)if 𝐽 has zero torsion, π”ͺ∩π”ͺξ…ž is a complex subalgebra of 𝔀.

Proof. (i) For any π‘βˆˆπ”€ one hasπ‘βˆˆπ”ͺ∩π”ͺξ…žβŸΊβˆƒπ‘‹,π‘Œβˆˆπ”€0,𝑍=π‘‹βˆ’π‘–π½π‘‹=π½π‘Œ+π‘–π‘ŒβŸΊβˆƒπ‘‹,π‘Œβˆˆπ”€0,𝑍=π‘‹βˆ’π‘–π½π‘‹,𝑋=𝐽Y,π‘Œ=βˆ’π½π‘‹βŸΊβˆƒπ‘‹βˆˆπ”€0,𝑍=π‘‹βˆ’π‘–π½π‘‹,𝑋=βˆ’π½2π‘‹ξ€·π½βŸΊβˆƒπ‘‹βˆˆker2+1,𝑍=𝑋.(2.7)
(ii) The result follows from Lemma 2.2(iii) since the first condition in (2.2) (with𝔭=𝔀0 and 𝐽𝔭=𝐽) is trivially satisfied and the second condition is the zero torsion condition.
(iii) From (ii), [π”ͺξ…ž,π”ͺξ…ž]βŠ‚π”ͺξ…ž, hence [π”ͺ,π”ͺξ…ž]βŠ‚π”ͺ and [π”ͺ∩π”ͺξ…ž,π”ͺ∩π”ͺξ…ž]βŠ‚π”ͺ∩π”ͺξ…ž. Clearly π”ͺ∩π”ͺξ…ž is stable by multiplication by 𝑖.

Definition 2.8. Let π½βˆΆπ”€0→𝔀0 be a zero torsion linear map. We say that 𝐽 is an extension of a 𝐢𝑅-structure if there exists a vector subspace 𝔭≠{0} of 𝔀0 such that 𝔭 equipped with the restriction 𝐽𝔭 of 𝐽 to 𝔭 is a 𝐢𝑅-structure on 𝔀0.

Definition 2.9. A real form 𝔲 of 𝔀 is said to be of type I (with respect to 𝔀0) if 𝔀0βˆ©π”²β‰ {0}.  𝔀0 is said to be type I if any real form 𝔲 of 𝔀 is of type I.

Remark 2.10. Introduce the real linear projections πœ‹1βˆΆπ”€β†’π”€0, πœ‹2βˆΆπ”€β†’π”€0 defined by 𝑧=πœ‹1(𝑧)+π‘–πœ‹2(𝑧) for π‘§βˆˆπ”€. Then a real form 𝔲 of 𝔀 is of type I if and only if kerπœ‹2βˆ£π”²β‰ {0}.

Proposition 2.11. Let π½βˆΆπ”€0→𝔀0 be a zero torsion linear map, π”ͺ={𝑋=π‘‹βˆ’π‘–π½π‘‹;π‘‹βˆˆπ”€0} and π”ͺξ…ž=𝑖π”ͺ.  𝐽 is an extension of a 𝐢𝑅-structure if and only if π”ͺ∩π”ͺξ…žβ‰ {0}.

Proof. From Lemma 2.7, π”ͺ∩π”ͺξ…ž is a complex subalgebra of 𝔀 and π”ͺ∩π”ͺξ…žξ‚={𝑋;π‘‹βˆˆker(𝐽2+1)}. If 𝐽 is an extension of a 𝐢𝑅-structure, one has {0}β‰ π”­βŠ‚ker(𝐽2+1) hence π”ͺ∩π”ͺξ…žβ‰ {0}. Conversely, if π”ͺ∩π”ͺξ…žβ‰ {0}, let 𝔭=ker(𝐽2+1). Then 𝔭 is stable under 𝐽, and if 𝐽𝔭 denotes the restriction of 𝐽 to 𝔭, conditions (2.3), (2.5) are trivially satisfied. Condition (2.4) holds true since, from Lemma 2.7(iii), π”ͺ∩π”ͺξ…ž is a subalgebra of 𝔀. Precisely, for 𝑋,π‘Œβˆˆπ”­,  𝑋,π‘Œβˆˆπ”ͺ∩π”ͺξ…ž hence [𝑖𝑋,π‘Œ]=[𝐽𝔭𝑋+𝑖𝑋,π‘Œβˆ’π‘–π½π”­π‘Œ]∈π”ͺ∩π”ͺξ…ž and (2.4) follows.

Corollary 2.12. There is a one-to-one correspondence between nontype I real forms of 𝔀 and zero torsion linear maps π½βˆΆπ”€0→𝔀0 which are no extension of a 𝐢𝑅-structure.

Proof. Let π½βˆΆπ”€0→𝔀0 be a zero torsion linear map that is no extension of a 𝐢𝑅-structure. Then π”ͺ∩π”ͺξ…ž={0}, hence π”ͺξ…ž={𝐽𝑋+𝑖𝑋;π‘‹βˆˆπ”€0} is a real form of 𝔀 which is nontype I. Conversely, if 𝔲 is a nontype I real form of 𝔀, then πœ‹2(𝔲)=𝔀0 implies 𝔲={𝐽𝑋+𝑖𝑋;π‘‹βˆˆπ”€0} for some linear map π½βˆΆπ”€0→𝔀0, that is, 𝔲=π”ͺξ…ž. As π”ͺξ…ž is a real subalgebra, 𝐽 has zero torsion from Lemma 2.7(ii). Now π”ͺ∩π”ͺξ…ž={0} since 𝔲 is a real form hence 𝐽 is not an extension of a 𝐢𝑅-structure.

Corollary 2.13. If 𝔀0 is of type I, then any zero torsion linear map π½βˆΆπ”€0→𝔀0 is an extension of a 𝐢𝑅-structure.

Proposition 2.14. Let 𝔲 be any real form of 𝔀,β€‰β€‰πœ,β€‰β€‰πœŽ the conjugations with respect to 𝔲,  𝔀0, and 𝑁=𝜎𝜏∈Aut𝔀. If 𝑁 has a nonzero fixed point, then 𝔲 is type I.

Proof. Let 𝑍 be a fixed point of 𝑁. 𝑁𝑍=𝑍 reads πœŽπ‘=πœπ‘. Consider 𝑉=πœŽπ‘=πœπ‘. Then πœŽπ‘‰=πœπ‘‰=𝑍. Hence π‘Š=𝑉+𝑍 has πœπ‘Š=𝑍+πœπ‘=𝑍+𝑉=π‘Š and similarly πœŽπ‘Š=π‘Š. Hence π‘Šβˆˆπ”€0βˆ©π”². Now, π‘Š=0 if and only if πœŽπ‘=πœπ‘=βˆ’π‘, that is, π‘–π‘βˆˆπ”€0βˆ©π”².

Corollary 2.15. If 𝔀0 is nonsolvable, then it is type I.

Proof. If 𝔀0 is nonsolvable, so is 𝔀. Now, it is known that any automorphism of a nonsolvable Lie algebra over a characteristic 0 field has a nonzero fixed point ([9]). Hence any real form 𝔲 of 𝔀 is type I.

3. Case of 𝔰𝔩(2,ℝ)

𝐺=𝑆𝐿(2,ℝ) denotes the Lie group of real 2Γ—2 matrices with determinant 1ξƒͺ𝜎=π‘Žπ‘π‘π‘‘,π‘Žπ‘‘βˆ’π‘π‘=1.(3.1) Its Lie algebra 𝔀0=𝔰𝔩(2,ℝ) consists of the zero trace real 2Γ—2 matrices ξƒͺ𝑋=π‘₯π‘¦π‘§βˆ’π‘₯=π‘₯𝐻+𝑦𝑋++π‘§π‘‹βˆ’,(3.2) with basis 𝐻=100βˆ’1ξ€Έ, 𝑋+=ξ€·0100ξ€Έ, π‘‹βˆ’=ξ€·0010ξ€Έ and commutation relations𝐻,𝑋+ξ€»=2𝑋+,𝐻,π‘‹βˆ’ξ€»=βˆ’2π‘‹βˆ’,𝑋+,π‘‹βˆ’ξ€»=𝐻.(3.3) Beside the basis (𝐻, 𝑋+, π‘‹βˆ’), we will also make use of the basis (π‘Œ1,π‘Œ2,π‘Œ3) where π‘Œ1=(1/2)𝐻,β€‰β€‰π‘Œ2=(1/2)(𝑋+βˆ’π‘‹βˆ’),β€‰β€‰π‘Œ3=(1/2)(𝑋++π‘‹βˆ’), with commutation relationsξ€Ίπ‘Œ1,π‘Œ2ξ€»=π‘Œ3,ξ€Ίπ‘Œ1,π‘Œ3ξ€»=π‘Œ2,ξ€Ίπ‘Œ2,π‘Œ3ξ€»=π‘Œ1.(3.4) The adjoint representation of 𝐺 on 𝔀0 is given by Ad(𝜎)𝑋=πœŽπ‘‹πœŽβˆ’1. The matrix Ξ¦ of Ad(𝜎) (𝜎 as in (3.1)) in the basis (𝐻,𝑋+,π‘‹βˆ’) isβŽ›βŽœβŽœβŽœβŽΞ¦=1+2π‘π‘βˆ’π‘Žπ‘π‘π‘‘βˆ’2π‘Žπ‘π‘Ž2βˆ’π‘22π‘π‘‘βˆ’π‘2𝑑2⎞⎟⎟⎟⎠.(3.5) The adjoint group Ad(𝐺) is the identity component of Aut  𝔀0 and one hasAut𝔀0=Ad(𝐺)βˆͺΞ¨0Ad(𝐺),Ξ¨0=diag(1,βˆ’1,βˆ’1).(3.6) The adjoint action of 𝐺 on 𝔀0 preserves the form π‘₯2+𝑦𝑧. The orbits are as follows: (i)the trivial orbit {0};(ii)the upper sheet 𝑧>0 of the cone π‘₯2+𝑦𝑧=0 (orbit of π‘‹βˆ’); (iii)the lower sheet 𝑧<0 of the cone π‘₯2+𝑦𝑧=0 (orbit of βˆ’π‘‹βˆ’); (iv)for all 𝑠>0 the one-sheet hyperboloid π‘₯2+𝑦𝑧=𝑠2 (orbit of 𝑠𝐻); (v)for all 𝑠>0 the upper sheet 𝑧>0 of the hyperboloid π‘₯2+𝑦𝑧=βˆ’π‘ 2 (orbit of 𝑠(βˆ’π‘‹++π‘‹βˆ’)); (vi)for all 𝑠>0 the lower sheet 𝑧<0 of the hyperboloid π‘₯2+𝑦𝑧=βˆ’π‘ 2 (orbit of 𝑠(𝑋+βˆ’π‘‹βˆ’)).

The orbits of 𝔀0 under the whole Aut𝔀0 are as follows, beside {0}: (I)the cone π‘₯2+𝑦𝑧=0 (orbit of π‘‹βˆ’); (II)the one-sheet hyperboloid π‘₯2+𝑦𝑧=𝑠2 (orbit of 𝑠H) (𝑠>0); (III) the two-sheet hyperboloid π‘₯2+𝑦𝑧=βˆ’π‘ 2 (orbit of 𝑠(𝑋+βˆ’π‘‹βˆ’)) (𝑠>0).

Proposition 3.1. Let 𝔀0=𝔰𝔩(2,ℝ), and π½βˆΆπ”€0→𝔀0 any linear map. 𝐽 has zero torsion if and only if it is equivalent to the endomorphism defined in the basis (π‘Œ1,π‘Œ2,π‘Œ3)(resp., (𝐻,𝑋+,π‘‹βˆ’)) by π½βˆ—βŽ›βŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽ (πœ†)=00βˆ’10πœ†0100,πœ†βˆˆβ„,(3.7)π½βˆ—(πœ†)β‰’π½βˆ—(πœ‡) for πœ†β‰ πœ‡ (resp., βŽ›βŽœβŽœβŽœβŽ1𝐽(𝛼)=0βˆ’2βˆ’12⎞⎟⎟⎟⎠1π›Όβˆ’π›Ό1βˆ’π›Όπ›Ό,π›Όβˆˆβ„,(3.8)𝐽(𝛼)≒𝐽(𝛽) for 𝛼≠𝛽).

Proof. Let 𝐽=(πœ‰π‘–π‘—)1⩽𝑖,𝑗⩽3 in the basis (𝐻,𝑋+,π‘‹βˆ’). The 9 torsion equations are in the basis (𝐻,𝑋+,π‘‹βˆ’): ξ€·πœ‰12∣1222+πœ‰11ξ€Έπœ‰12+ξ€·πœ‰22βˆ’πœ‰11ξ€Έπœ‰31βˆ’ξ€·πœ‰21+2πœ‰13ξ€Έπœ‰32ξ‚€πœ‰=0,12∣2221πœ‰12ξ€·πœ‰+1+22ξ€Έ2ξ‚βˆ’πœ‰31πœ‰21βˆ’2πœ‰32πœ‰23ξ€·πœ‰=0,12∣331+2πœ‰12ξ€Έπœ‰31ξ€·πœ‰βˆ’222+2πœ‰11ξ€Έπœ‰32+2πœ‰33πœ‰32ξ€·πœ‰=0,13∣121βˆ’2πœ‰13ξ€Έπœ‰11+2πœ‰23πœ‰12+πœ‰31πœ‰23βˆ’ξ€·πœ‰21+2πœ‰13ξ€Έπœ‰33ξ€·πœ‰=0,13∣2222βˆ’2πœ‰11ξ€Έπœ‰23+ξ€·πœ‰21+2πœ‰13ξ€Έπœ‰21βˆ’2πœ‰33πœ‰23=0,13∣3πœ‰31πœ‰21βˆ’2πœ‰31πœ‰13βˆ’2+2πœ‰32πœ‰23ξ€·πœ‰βˆ’233ξ€Έ2=0,23∣14πœ‰13πœ‰12βˆ’1βˆ’πœ‰22πœ‰11βˆ’πœ‰32πœ‰23+ξ€·πœ‰22βˆ’πœ‰11ξ€Έπœ‰33=0,23∣24πœ‰23πœ‰12βˆ’ξ€·πœ‰22+πœ‰33ξ€Έπœ‰21=0,23∣34πœ‰32πœ‰13βˆ’ξ€·πœ‰22+πœ‰33ξ€Έπœ‰31=0.(3.9)𝐽 has at least one real eigenvalue πœ†. Let π‘£βˆˆπ”€0,  𝑣≠0, an eigenvector associated to πœ†. From the classification of the Aut𝔀0 orbits of 𝔀0, we then get 3 cases according to whether 𝑣 is on the orbit (I), (II), (III)  (in the cases (II), (III) one may choose 𝑣 so that 𝑠=1). Case 1. There exists πœ‘βˆˆAut𝔀0 such that 𝑣=πœ‘(π‘‹βˆ’). Then, replacing 𝐽 by πœ‘βˆ’1π½πœ‘, we may suppose πœ‰13=πœ‰23=0. That case is impossible from 13∣2 and 13∣3.Case 2. There exists πœ‘βˆˆAut𝔀0 such that 𝑣=πœ‘(𝐻). Then we may suppose πœ‰21=πœ‰31=0. Then from 12∣2, πœ‰23πœ‰32β‰ 0, and 23∣2, 23∣3 yield πœ‰12=πœ‰13=0. Then 12∣3 and 13∣2 successively give πœ‰33=πœ‰22+2πœ‰11 and πœ‰11=0. Now 12∣2 and 23∣1 read respectively βˆ’πœ‰23πœ‰32+(πœ‰22)2+1=0, and πœ‰23πœ‰32βˆ’(πœ‰22)2+1=0. Hence that case is impossible.Case 3. There exists πœ‘βˆˆAut𝔀0 such that 𝑣=πœ‘(𝑋+βˆ’π‘‹βˆ’). Then we may suppose that 𝑣=𝑋+βˆ’π‘‹βˆ’. Now instead of the basis (𝐻,𝑋+,π‘‹βˆ’), we consider the basis (π‘Œ1,π‘Œ2,π‘Œ3). The matrix of 𝐽 in the basis (π‘Œ1,π‘Œ2,π‘Œ3) has the formπ½βˆ—=βŽ›βŽœβŽœβŽœβŽπœ‚110πœ‚13πœ‚21πœ†πœ‚23πœ‚310πœ‚23⎞⎟⎟⎟⎠.(3.10) Then the 9 torsion equations βˆ—π‘–π‘—βˆ£π‘˜ (the star is to underline that the new basis is in use) for 𝐽 in that basis are as follows: ξ€·πœ‚βˆ—12∣131+πœ‚13ξ€Έξ€·πœ‚πœ†βˆ’31βˆ’πœ‚13ξ€Έπœ‚11ξ€·πœ‚=0,βˆ—12∣211ξ€Έπœ‚+πœ†23βˆ’πœ‚21πœ‚31=0,βˆ—12∣3πœ‚11ξ€·πœ‚πœ†βˆ’1+31ξ€Έ2βˆ’ξ€·πœ‚11ξ€Έπœ‚+πœ†33=0,βˆ—13∣1πœ‚23πœ‚13+πœ‚21πœ‚11+πœ‚23πœ‚31βˆ’πœ‚21πœ‚33=0,βˆ—13∣2πœ‚11ξ€·πœ‚πœ†+1+21ξ€Έ2+ξ€·πœ‚23ξ€Έ2+πœ‚31πœ‚13βˆ’ξ€·πœ‚11ξ€Έπœ‚βˆ’πœ†33=0,βˆ—13∣3πœ‚23πœ‚11βˆ’πœ‚21ξ€·πœ‚13+πœ‚31ξ€Έβˆ’πœ‚23πœ‚33=0,βˆ—23∣1πœ‚11ξ€·πœ‚πœ†+1βˆ’13ξ€Έ2+ξ€·πœ‚11ξ€Έπœ‚βˆ’πœ†33=0,βˆ—23∣2πœ‚23πœ‚13βˆ’ξ€·πœ‚33ξ€Έπœ‚+πœ†21ξ€·πœ‚=0,βˆ—23∣331+πœ‚13ξ€Έξ€·πœ‚πœ†+31βˆ’πœ‚13ξ€Έπœ‚33=0.(3.11) From βˆ—12∣1 and βˆ—23∣3, πœ‚11ξ€·πœ‚31βˆ’πœ‚13ξ€Έ=βˆ’πœ‚33ξ€·πœ‚31βˆ’πœ‚13ξ€Έ.(3.12)Case 1. Suppose first that πœ‚31=πœ‚13. Then πœ†πœ‚31=0.Subcase 1. Consider the subcase πœ‚31=0.βˆ—13∣1 and βˆ—13∣3 read, respectively, (πœ‚33βˆ’πœ‚11)πœ‚21=0,(πœ‚33βˆ’πœ‚11)πœ‚23=0. Suppose πœ‚33β‰ πœ‚11. Then πœ‚21=πœ‚23=0, and βˆ—13∣2 gives πœ‚11πœ†+1=(πœ‚11βˆ’πœ†)πœ‚33, which implies πœ‚33=0 by βˆ—23∣1. As βˆ—12∣3 then reads 1=0, this case πœ‚33β‰ πœ‚11 is not possible. Now, the case πœ‚33=πœ‚11 is not possible either since then βˆ—23∣1 would read (πœ‚11)2+1=0. We conclude that the Subcase 1 is not possible. Subcase 2. Hence we are in the Subcase 2:πœ‚31β‰ 0. Then πœ†=0. From βˆ—13∣2, πœ‚33πœ‚11β‰ 0. Then βˆ—23∣1 yields πœ‚33=(βˆ’1+(πœ‚31)2)/πœ‚11 and βˆ—13∣2 reads (πœ‚21)2+(πœ‚23)2+2=0. This Subcase 2 is not possible either.Case 2. Hence Case 1 is not possible, and we are necessarily in the Case 2:πœ‚31β‰ πœ‚13. From (3.12), πœ‚33=βˆ’πœ‚11. Then βˆ—13∣2 reads (πœ‚11)2+(πœ‚21)2+(πœ‚23)2+1+πœ‚31πœ‚13=0 hence πœ‚31β‰ 0 and πœ‚13=βˆ’((πœ‚11)2+(πœ‚21)2+(πœ‚23)2+1)/πœ‚31. From βˆ—12∣2,πœ‚21=(πœ‚23(πœ‚11+πœ†))/πœ‚31. Then βˆ—23∣2 reads πœ‚23(((πœ‚23)2+πœ†2+1)(πœ‚31)2+(πœ‚11+πœ†)2(πœ‚23)2)=0, that is, πœ‚23=0, which implies πœ‚21=0. Now βˆ—12∣1 reads πœ†(1+(πœ‚11)2βˆ’(πœ‚31)2)=βˆ’πœ‚11(1+(πœ‚11)2+(πœ‚31)2). The subcase πœ‚11β‰ 0 is not possible since then βˆ—12∣3 would yield πœ†=βˆ’((πœ‚11)2+(πœ‚31)2βˆ’1)/2πœ‚11 and βˆ—12∣1 would read ((πœ‚11)2+(πœ‚31+1)2)((πœ‚11)2+(πœ‚31βˆ’1)2)=0. Hence πœ‚11=0. Then βˆ—12∣3 reads (πœ‰31)2=1. The condition (πœ‰31)2=1 now implies the vanishing of all the torsion equations. In that case π½βˆ—=βŽ›βŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽ 00βˆ’πœ€0πœ†0πœ€00,πœ€=Β±1.(3.13) Then in the basis (𝐻,𝑋+,π‘‹βˆ’)βŽ›βŽœβŽœβŽœβŽœβŽœβŽπœ€π½=0βˆ’2βˆ’πœ€2πœ€πœ†2βˆ’πœ†2πœ†πœ€βˆ’2πœ†2⎞⎟⎟⎟⎟⎟⎠.(3.14) The cases πœ€=Β±1 are equivalent under Ξ¨0.

Remark 3.2. π½βˆ—(πœ†) is an extension of a 𝐢𝑅-structure, in agreement with Corollary 2.15.

4. Complex Structures on 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ)

We consider the basis (π‘Œ1(1),π‘Œ2(1),π‘Œ3(1),π‘Œ1(2),π‘Œ2(2),π‘Œ3(2)) of 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ), with the upper index referring to the first or second factor. The automorphisms of 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ) fall into 2 kinds: the first kind is comprised of the diag(Ξ¦1,Ξ¦2), Ξ¦1,Ξ¦2∈Aut𝔰𝔩(2,ℝ), and the second kind is comprised of the Ξ“βˆ˜diag(Ξ¦1,Ξ¦2), with Ξ“ the switch between the two factors of 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ).

Proposition 4.1. Any integrable complex structure 𝐽 on 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ) is equivalent under some first kind automorphism to the endomorphism given in the basis (π‘Œ1(1),π‘Œ2(1),π‘Œ3(1),π‘Œ1(2),π‘Œ2(2),π‘Œ3(2)) by the matrix ξ‚π½βˆ—ξ€·πœ‰22,πœ‰25ξ€Έ=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽ00βˆ’10000πœ‰2200πœ‰250ξ€·πœ‰10000000000βˆ’10βˆ’22ξ€Έ2+1πœ‰2500βˆ’πœ‰220⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠000100,πœ‰22,πœ‰25βˆˆβ„,πœ‰25β‰ 0.(4.1)ξ‚π½βˆ—(πœ‰22,πœ‰25) is equivalent to ξ‚π½βˆ—(πœ‰ξ…ž22,πœ‰ξ…ž25) under some first (resp., second) kind automorphism if and only if πœ‰ξ…ž22=πœ‰22,β€‰πœ‰ξ…ž25=πœ‰25 (resp., πœ‰ξ…ž22=βˆ’πœ‰22,β€‰β€‰πœ‰ξ…ž25=βˆ’((πœ‰22)2+1)/πœ‰25).

Proof. Let 𝐽=(πœ‰π‘–π‘—)1⩽𝑖,𝑗⩽6=𝐽1𝐽2𝐽3𝐽4, (𝐽1,𝐽2,𝐽3,𝐽43Γ—3 blocks), an integrable complex structure in the basis (π‘Œβ„“(π‘˜)). From Proposition 3.1, with some first kind automorphism, one may suppose 𝐽1=ξ‚΅00βˆ’10πœ‰220100ξ‚Ά,  𝐽4=ξ‚΅00βˆ’10πœ‰550100ξ‚Ά. As Tr(𝐽)=0,β€‰β€‰πœ‰55=βˆ’πœ‰22. Then one is led to (4.1), and the result follows.

Remark 4.2. The complex subalgebra π”ͺ associated to ξ‚π½βˆ—(πœ‰22,πœ‰25) has basis ξ‚π‘Œ1(1)=π‘Œ1(1)βˆ’π‘–π‘Œ3(1), ξ‚π‘Œ1(2)=π‘Œ1(2)βˆ’π‘–π‘Œ3(2), ξ‚π‘Œ2(2)=βˆ’π‘–πœ‰25π‘Œ2(1)+(1+π‘–πœ‰22)π‘Œ2(2). The complexification 𝔰𝔩(2)×𝔰𝔩(2) of 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ) has weight spaces decomposition with respect to the Cartan subalgebra π”₯=β„‚π‘Œ2(1)βŠ•β„‚π‘Œ2(2): ξ‚€π‘Œπ”₯βŠ•β„‚1(1)+π‘–π‘Œ3(1)ξ‚ξ‚€π‘ŒβŠ•β„‚1(2)+π‘–π‘Œ3(2)ξ‚ξ‚π‘ŒβŠ•β„‚1(1)ξ‚π‘ŒβŠ•β„‚1(2).(4.2) Then ξ‚π‘Œπ”ͺ=(π”₯∩π”ͺ)βŠ•β„‚1(1)ξ‚π‘ŒβŠ•β„‚1(2) with ξ‚π‘Œπ”₯∩π”ͺ=β„‚2(2), which is a special case of the general fact proved in [10] that any complex (integrable) structure on a reductive Lie group of class I is regular.

5. Case of 𝔫

Let 𝔫 be the real 3-dimensional Heisenberg Lie algebra with basis (π‘₯1,π‘₯2,π‘₯3) and commutation relations [π‘₯1,π‘₯2]=π‘₯3.

Proposition 5.1. Let π½βˆΆπ”«β†’π”« any linear map. 𝐽 has zero torsion if and only if it is equivalent to one of the endomorphisms defined in the basis (π‘₯1,π‘₯2,π‘₯3) by the following: (i)π‘†ξ€·πœ‰33ξ€Έ=βŽ›βŽœβŽœβŽœβŽ0βˆ’1010000πœ‰33⎞⎟⎟⎟⎠,πœ‰33βˆˆβ„,(5.1)(ii)π·ξ€·πœ‰11ξ€Έ=βŽ›βŽœβŽœβŽœβŽœβŽœβŽπœ‰11000πœ‰110ξ€·πœ‰0011ξ€Έ2βˆ’12πœ‰11⎞⎟⎟⎟⎟⎟⎠,πœ‰11βˆˆβ„,πœ‰11β‰ 0,(5.2)(iii)βŽ›βŽœβŽœβŽœβŽœβŽπ‘‡(π‘Ž,𝑏)=0βˆ’π‘Žπ‘01𝑏000π‘Žπ‘βˆ’1π‘βŽžβŽŸβŽŸβŽŸβŽŸβŽ ,π‘Ž,π‘βˆˆβ„,𝑏≠0.(5.3) Any two distinct endomorphisms in the preceding list are nonequivalent. 𝑇(π‘Ž,𝑏) is equivalent to π‘‡ξ…žβŽ›βŽœβŽœβŽœβŽœβŽ(π‘Ž,𝑏)=π‘βˆ’π‘0π‘Ž0000π‘Žπ‘βˆ’1π‘βŽžβŽŸβŽŸβŽŸβŽŸβŽ .(5.4)

Proof. Let 𝐽=(πœ‰π‘–π‘—)1⩽𝑖,𝑗⩽3 in the basis (π‘₯1,π‘₯2,π‘₯3).The 9 torsion equations reduce to πœ‰13=πœ‰23=0 and equation 12∣3 (with the general notation introduced after Definition 2.6) which reads πœ‰33Tr(𝐴)=det(𝐴)βˆ’1,(5.5) where 𝐴=πœ‰11πœ‰12πœ‰21πœ‰22. Suppose first Tr(𝐴)=0. Then 𝐴2=βˆ’πΌ, so that 𝐴 is similar over β„‚, hence over ℝ, to ξ€·0βˆ’110ξ€Έ. Hence 𝐽≑0βˆ’10100βˆ—βˆ—πœ‰33ξ‚Ά. Now, since πœ‰33 does not belong to the spectrum of ξ€·0βˆ’110ξ€Έ, taking the automorphism ξ‚€100010𝛼𝛽1 of 𝔫 for suitable 𝛼,π›½βˆˆβ„, one gets 𝐽≑𝑆(πœ‰33). Suppose now Tr(𝐴)β‰ 0. Then πœ‰33=(det(𝐴)βˆ’1)/Tr(𝐴). If 𝐴 is a scalar matrix, that is, 𝐴=πœ‰11𝐼, then 𝐽=πœ‰11000πœ‰110βˆ—βˆ—((πœ‰11)2βˆ’1)/2πœ‰11ξƒͺ≑𝐷(πœ‰11). If 𝐴 is not a scalar matrix, then 𝐴 is similar to ξ€·0βˆ’π‘Žπ‘1𝑏 for some π‘Ž,π‘βˆˆβ„, and 𝑏≠0 from the trace. Then 𝐽≑𝑇(π‘Ž,𝑏). Finally, 𝑇′(π‘Ž,𝑏)≑𝑇(π‘Ž,𝑏) since the matrices ξ€·0βˆ’π‘Žπ‘1𝑏 and ξ€·π‘βˆ’π‘π‘Ž0ξ€Έ are similar for they have the same spectrum and are no scalar matrices.

Remark 5.2. 𝑆(πœ‰33) is an extension of a rank 1 𝐢𝑅-structure; however, 𝐷(πœ‰11),𝑇(π‘Ž,𝑏) are not.

6. 𝐢𝑅-Structures on 𝔫

Proposition 6.1. (i) Any linear map π½βˆΆπ”«β†’π”« which has zero torsion and is an extension of a rank 1 𝐢𝑅-structure on 𝔫 such that 𝔭 is nonabelian is equivalent to a unique βŽ›βŽœβŽœβŽœβŽ0βˆ’1010000πœ‰33⎞⎟⎟⎟⎠,πœ‰33βˆˆβ„.(6.1)
(ii) Any linear map π½βˆΆπ”«β†’π”« which is an extension of a rank 1 𝐢𝑅-structure on 𝔫 such that 𝔭 is abelian is equivalent to a unique βŽ›βŽœβŽœβŽœβŽπœ‰11⎞⎟⎟⎟⎠000010βˆ’10,πœ‰11βˆˆβ„.(6.2)𝐽 has nonzero torsion.

Proof. For any nonzero π‘‹βˆˆπ”­, (𝑋,𝐽𝔭𝑋) is a basis of 𝔭. In case (i), [𝑋,𝐽𝔭𝑋]β‰ 0, since 𝔭 is nonabelian. Then [𝑋,𝐽𝔭𝑋]=πœ‡π‘₯3, πœ‡β‰ 0, and π‘₯3βˆ‰π”­ since otherwise 𝔭 would be abelian. One may extend 𝐽𝔭 to 𝔫 in the basis (𝑋,𝐽𝔭𝑋,πœ‡π‘₯3) as βŽ›βŽœβŽœβŽœβŽπ½=0βˆ’1πœ‰1310πœ‰2300πœ‰33⎞⎟⎟⎟⎠,(6.3) and 𝐽 has zero torsion only if πœ‰13=πœ‰23=0. In case (ii), necessarily π‘₯3βˆˆπ”­ since 𝔭 is abelian. Hence (π‘₯3,𝐽𝔭π‘₯3) is a basis for 𝔭. Take any linear extension 𝐽 of 𝐽𝔭 to 𝔫. There exists some eigenvector 𝑦1β‰ 0 of 𝐽 associated to some eigenvalue πœ‰11βˆˆβ„. Then 𝑦1βˆ‰π”­, which implies [𝑦1,𝐽π‘₯3]β‰ 0, for otherwise 𝑦1 would commute to the whole of 𝔫 and then be some multiple of π‘₯3βˆˆπ”­. Hence [𝑦1,𝐽π‘₯3]=πœ†π‘₯3, πœ†β‰ 0, and dividing 𝑦1 by πœ† one may suppose πœ†=1. In the basis 𝑦1,𝑦2=𝐽π‘₯3,𝑦3=π‘₯3 one has βŽ›βŽœβŽœβŽœβŽπœ‰π½=11⎞⎟⎟⎟⎠000010βˆ’10,(6.4) and (ii) follows.

Remark 6.2. Let 𝑁={[π‘₯,𝑦,𝑧];π‘₯,𝑦,π‘§βˆˆβ„} denote the Heisenberg group, where ξ‚€[π‘₯,𝑦,𝑧]=1π‘₯𝑧01𝑦001. 𝑁 can also be realized ([11, 12]) as the boundary 𝑀2={(𝜁,𝑀)βˆˆβ„‚2;β„‘πœ=𝑀𝑀} of the Siegel half-space equipped with the multiplication (𝜁1,𝑀1)(𝜁2,𝑀2)=(𝜁1+𝜁2+2𝑖𝑀1𝑀2,𝑀1+𝑀2). The map Ξ¨βˆΆπ‘β†’π‘€2 defined by Ξ¨([π‘₯,𝑦,𝑧])=(π‘§βˆ’(1/2)π‘₯𝑦+𝑖(π‘₯2+𝑦2)/4,(1/2)(π‘₯βˆ’π‘–π‘¦)) is an isomorphism. If 𝑃, 𝑄 denote the left invariant vector fields associated, respectively, to π‘₯1,π‘₯2, then (𝑑Ψ)(𝑃+𝑖𝑄)=2𝑖𝑀(πœ•/πœ•πœ)+(πœ•/πœ•π‘€), hence the left invariant 𝐢𝑅-structure on 𝑁 associated to the 𝐢𝑅-structure on 𝔫 introduced in (i) is the 𝐢𝑅-structure on 𝑀2 induced from β„‚2. The left invariant 𝐢𝑅-structure on 𝑁 associated to the 𝐢𝑅-structure on 𝔫 introduced in (ii) is not 𝐢𝑅-diffeomorphic to the 𝐢𝑅-structure on 𝑀2 induced from β„‚2, since the former has zero Levi form.

7. Complex Structures on 𝔫×𝔫

We will use for commutation relations [π‘₯1,π‘₯2]=π‘₯5,[π‘₯3,π‘₯4]=π‘₯6. The automorphisms of 𝔫×𝔫 fall into 2 kinds. The first kind is comprised of the matrices βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘Ξ¦=11𝑏12𝑏000021𝑏22000000𝑏33𝑏340000𝑏43𝑏4400𝑏51𝑏52𝑏53𝑏54𝑏11𝑏22βˆ’π‘12𝑏210𝑏61𝑏62𝑏63𝑏640𝑏33𝑏44βˆ’π‘34𝑏43⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑏11𝑏22βˆ’π‘12𝑏21𝑏33𝑏44βˆ’π‘34𝑏43ξ€Έβ‰ 0.(7.1) The second kind ones are Ξ¨=ΘΦ where Ξ¦ is first kind and βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽΞ˜=001000000100100000010000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠000001000010.(7.2)

Proposition 7.1. Any integrable complex structure 𝐽 on 𝔫×𝔫 is equivalent under some first kind automorphism to one of the following: (i)ξ‚π‘†πœ€ξ€·πœ‰55ξ€Έ=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽ0βˆ’10000100000000βˆ’1000010000000πœ‰55ξ‚€ξ€·πœ‰βˆ’πœ€55ξ€Έ2+10000πœ€βˆ’πœ‰55⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,πœ€=Β±1,πœ‰55βˆˆβ„,(7.3)ξ‚π‘†πœ€β€²(πœ‰ξ…ž55) is equivalent to ξ‚π‘†πœ€(πœ‰55) (πœ€,πœ€ξ…ž=Β±1;πœ‰ξ…ž55,πœ‰55βˆˆβ„) under some first (resp., second) kind automorphism if and only if πœ€ξ…ž=πœ€,πœ‰ξ…ž55=πœ‰55 (resp.,β€‰β€‰πœ€ξ…ž=βˆ’πœ€,πœ‰ξ…ž55=βˆ’πœ‰55 ).(ii)ξ‚π·ξ€·πœ‰11ξ€Έ=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπœ‰11ξ‚€ξ€·πœ‰0βˆ’11ξ€Έ2+10000πœ‰11ξ‚€ξ€·πœ‰0βˆ’11ξ€Έ2+10010βˆ’πœ‰11000010βˆ’πœ‰11ξ€·πœ‰00000011ξ€Έ2βˆ’12πœ‰11βˆ’ξ‚€ξ€·πœ‰11ξ€Έ2+122πœ‰11100002πœ‰11ξ€·πœ‰1βˆ’11ξ€Έ22πœ‰11⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,πœ‰11βˆˆβ„β§΅{0},(7.4)𝐷(πœ‰ξ…ž11) is equivalent to 𝐷(πœ‰11) (πœ‰ξ…ž11,πœ‰11βˆˆβ„) under some first (resp., second) kind automorphism if and only if πœ‰ξ…ž11=πœ‰11 (resp., πœ‰ξ…ž11=βˆ’πœ‰11 ),(iii)ξ‚π‘‡ξ€·πœ‰33,πœ‰43ξ€Έ=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽ0βˆ’πœ‰43πœ‰33βˆ’πœ‰43πœ‰33πœ‰43πœ‰33βˆ’1001βˆ’πœ‰33βˆ’ξ€·πœ‰33ξ€Έ2+1βˆ’πœ‰43πœ‰33πœ‰33πœ‰33000πœ‰33πœ‰33βˆ’πœ‰330010πœ‰43πœ‰0000000βˆ’43πœ‰33βˆ’1πœ‰33βˆ’ξ€·πœ‰43πœ‰33ξ€Έπœ‰βˆ’243πœ‰33+ξ€·πœ‰33ξ€Έ2+1ξ€·πœ‰33ξ€Έ2πœ‰0000143πœ‰33βˆ’1πœ‰33⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,πœ‰33βˆˆβ„β§΅{0},πœ‰43βˆˆβ„,(7.5)𝑇(πœ‰ξ…ž33,πœ‰ξ…ž43) is equivalent to 𝑇(πœ‰33,πœ‰43) (πœ‰ξ…ž33,πœ‰33βˆˆβ„β§΅{0},πœ‰ξ…ž43,πœ‰43βˆˆβ„) under some first (resp., second) kind automorphism if and only if πœ‰ξ…ž33=πœ‰33,πœ‰ξ…ž43=πœ‰43 (resp. β€‰πœ‰ξ…ž33=βˆ’πœ‰33,πœ‰ξ…ž43=βˆ’πœ‰43).
Finally, the cases (i), (ii), (iii) are mutually nonequivalent, either under first or second kind automorphism.

Proof. Let 𝐽=(πœ‰π‘–π‘—)1⩽𝑖,𝑗⩽6 an integrable complex structure in the basis (π‘₯π‘˜)1β©½π‘˜β©½6. Denote 𝐽1=ξ‚€πœ‰11πœ‰12πœ‰21πœ‰22, 𝐽2=ξ‚€πœ‰13πœ‰14πœ‰23πœ‰24, 𝐽3=ξ‚€πœ‰31πœ‰32πœ‰41πœ‰42, 𝐽4=ξ‚€πœ‰33πœ‰34πœ‰43πœ‰44. Then π½βˆ—1=ξƒ©πœ‰11πœ‰12πœ‰15πœ‰21πœ‰22πœ‰25πœ‰51πœ‰52πœ‰55ξƒͺ and π½βˆ—3=ξƒ©πœ‰33πœ‰34πœ‰36πœ‰43πœ‰44πœ‰46πœ‰63πœ‰64πœ‰66ξƒͺ are zero torsion linear maps from 𝔫 to 𝔫, hence equivalent to type (5.1), (5.2), or (5.3) in Proposition 5.1. It can be checked that their being different types would contradict with 𝐽2=βˆ’1. Hence, modulo equivalence under some first kind automorphism, we get 3 cases: Case 1. π½βˆ—1=ξ‚΅0βˆ’1010000πœ‰55ξ‚Ά, π½βˆ—3=ξ‚΅0βˆ’1010000πœ‰66ξ‚Ά,Case 2. π½βˆ—1=𝐷(πœ‰11),π½βˆ—3=𝐷(πœ‰33), (πœ‰11,πœ‰33β‰ 0), andCase 3. π½βˆ—1=0πœ‰1201πœ‰22000πœ‰55ξƒͺ, π½βˆ—3=ξƒ©πœ‰33βˆ’πœ‰330πœ‰430000πœ‰66ξƒͺ, (πœ‰22,πœ‰33β‰ 0).Case 1 (resp., 2 and 3) leads to (7.3) (resp., (7.4), (7.5)). The assertion about equivalence in Cases 1 and 2 are readily proved, as is equivalence under some first kind automorphism in Case 3 and the nonequivalence of the 3 types. Consider now Ξ˜ξ‚π‘‡(πœ‰33,πœ‰43)Ξ˜βˆ’1. It is equivalent under some first kind automorphism to some 𝑇(πœ‚33,πœ‚43). That implies that the matrices ξ‚€πœ‰33βˆ’πœ‰33πœ‰430, ξ‚€0βˆ’πœ‚43πœ‚331βˆ’πœ‚33 are similar, which amounts to their having same trace and same determinant, that is, πœ‚33=βˆ’πœ‰33,πœ‚43=βˆ’πœ‰43. As 𝑇(πœ‰ξ…ž33,πœ‰ξ…ž43) is equivalent to 𝑇(πœ‰33,πœ‰43) under some second kind automorphism if and only if it is equivalent to Ξ˜ξ‚π‘‡(πœ‰33,πœ‰43)Ξ˜βˆ’1 under some first kind automorphism, the assertion about second kind equivalence in Case 3 follows.

Remark 7.2. In Case 3, had we used π½βˆ—3=0πœ‰3401πœ‰44000πœ‰66ξƒͺ, then we would have to separate further into 2 subcases: subcase πœ‰12β‰ 0, ξ‚π‘‡ξ€·πœ‰12,πœ‰22ξ€Έ=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽ0πœ‰12βˆ’πœ‰22πœ‰12βˆ’ξ€·πœ‰12ξ€Έ+1001πœ‰22πœ‰12+1πœ‰12βˆ’πœ‰22000βˆ’πœ‰120πœ‰12001πœ‰221βˆ’πœ‰22πœ‰000000βˆ’12+1πœ‰22βˆ’ξ€·πœ‰22ξ€Έ2+ξ€·πœ‰12ξ€Έ+12πœ‰22πœ‰12πœ‰000012πœ‰22πœ‰12+1πœ‰22⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,πœ‰12πœ‰22β‰ 0,(7.6) subcase πœ‰12=0, ξ‚π‘‡ξ€·πœ‰22ξ€Έ=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽ00βˆ’10001πœ‰220100100000βˆ’πœ‰22βˆ’ξ‚€ξ€·πœ‰22ξ€Έ2+11βˆ’πœ‰221000000βˆ’πœ‰221πœ‰22ξ€·πœ‰0000βˆ’22ξ€Έ2+1πœ‰221πœ‰22⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,πœ‰22β‰ 0.(7.7)

Remark 7.3. ξ‚π‘†πœ€(πœ‰55) is abelian (i.e., the corresponding complex subalgebra π”ͺ is abelian).

Remark 7.4. If one looks for zero torsion linear maps instead of complex structures, then π½βˆ—1 and π½βˆ—3 may be of different types.