Let V connected simply connected 2n-dimensional real vector space. The so-called complex structure J on V is a linear transformation J : V → V , satisfying:

J 2 = − id : V → V .

Let M be a 2n-dimensional smooth manifold, and J be a smooth (1,1)-type tensor field on M. For each point x ∈ M , J_k a linear transformation from the tangent space T x M to itself. If each J k ( x ) ( x ∈ M ) is a complex structure on the tangent space T x M , then the tensor field J is called a almost complex structure on M. The smoothness of the tensor field J implies that if X is a smooth tangent vector field on M, then JX is also a smooth tangent vector field on M.

Let G be a Lie group with a left-invariant almost complex structure, g = L i e G . Then, we can define a linear map J : g → g and J 2 = − id . J is called a complex structure on g. If J satisfies:

[ J X , J Y ] = [ X , Y ] + J [ J X , Y ] + J [ X , J Y ] for any ( X , Y ∈ g ) , (1)

then J is integrable. Without distinction, the left-invariant integrable almost complex structure on G and the integrable complex structure on g are both denoted by J.

Let g be a Lie algebra. Suppose

g 0 = g , g 1 = [ g 0 , g ] , ⋯ , g l = [ g l − 1 , g ] , ⋯ (2)

It can be easily proven that g i is an ideal of g, and g i ⊆ g i − 1 . The sequence { g k , k ≥ 0 } is called the descending central series of g. If there exists an s ∈ N such that g s = { 0 } and g s − 1 ≠ { 0 } , then g is called an s-step nilpotent Lie algebra [5] [6] .

Let G be a 2n-dimensional real nilpotent Lie group with a left-invariant integrable almost complex structure, g = L i e G , and g * be the dual space of g. Let { w 1 , w 2 , ⋯ , w n } denote a complex basis, and { w 1 , w ¯ 1 , w 2 , w ¯ 2 , ⋯ , w n , w ¯ n } denote a corresponding real basis. Therefore,

d w i = ∑ j < k A i j k w j ∧ w k + ∑ j , k B i j k w j ∧ w ¯ k + ∑ j < k C i j k w ¯ j ∧ w ¯ k ( 1 ≤ i ≤ n ) , (3)

because d α ( X , Y ) = − α ( [ X , Y ] ) ( X , Y ∈ g C , α ∈ ( g C ) * ) , we use the exterior derivative on g * to describe the Lie bracket on g.

Property 1 [7] . Let G be a real nilpotent Lie group, g = L i e G . G has a left-invariant integrable almost complex structure if and only if C i j k = 0 , i.e.

d w i = ∑ j < k ≤ n A i j k w j ∧ w k + ∑ j , k ≤ n B i j k w j ∧ w ¯ k ( 1 ≤ i ≤ n ). (4)

The structure equation can define connected and simply connected nilpotent Lie groups left-invariant integrable almost complex structure, so we can study some properties of Lie groups through this structure equation.

Definition 1 [8] . Let G be a connected simply connected s-step nilpotent Lie group, g = L i e G . Define a sequence in g as

g 0 = 0 , g 1 = { X ∈ g | [ X , g ] ⊆ g 0 } , ⋯ , g k = { X ∈ g | [ X , g ] ⊆ g k − 1 } , ⋯ . (5)

Then g₁ is the center of g, where g 1 ≠ { 0 } . g s = g , g k ⊆ g k + 1 , and for any k ( 0 ≤ k ≤ s − 1 ) chosen such that dim g k < dim g k + 1 . Thus, there exists an ascending central series

g 0 = 0 ⊂ g 1 ⊂ g 2 ⊂ ⋯ ⊂ g s − 1 ⊂ g s = g . (6)

Property 2.If the sequence { g k , k ≥ 0 } satisfies Equation (5), then:

a) g₁ is the center of g and g 1 ≠ 0 ;

b) For any k ≥ 0 , g k ⊆ g k + 1 ;

c) For any k ( 0 ≤ k ≤ s − 1 ) such that dim g k < dim g k + 1 ( 0 ≤ k ≤ s − 1 ) . We have g 0 = 0 ⊂ g 1 ⊂ g 2 ⊂ ⋯ ⊂ g s − 1 ⊂ g s = g .

Proof. a) Since g 0 = { 0 } , then g 1 = { X ∈ g | [ X , g ] = 0 } = C ( g ) . Moreover, since g is a nilpotent Lie algebra, C ( g ) ≠ 0 , hence g₁ is the center of g and g 0 ⊂ g 1 .

b) We use induction to prove g k ⊆ g k + 1 ( k ≥ 0 ) .

For k = 0 , by a), we have g 0 ⊂ g 1 . Assume that when k = i holds, g i ⊆ g i + 1 . We’ll prove that for k = i + 1 , g i + 1 ⊆ g i + 2 . For any x 1 ∈ g i + 1 and x ∈ g , we have [ x 1 , x ] ∈ g i ⊆ g i + 1 . According to Equation (5), x 1 ∈ g i + 2 , thus g i + 1 ⊆ g i + 2 , hence g k ⊆ g k + 1 ( k ≥ 0 ) .

c) First, we’ll prove that there exists an integer s such that g s = g .

Since g is an s-step nilpotent Lie algebra, there exists a descending central series

g 0 = g ⊃ g 1 = [ g , g 0 ] ⊃ ⋯ ⊃ g k + 1 = [ g , g k ] ⊃ ⋯ g s = { 0 } .

Next, we’ll use induction to prove g s − i ⊆ g i . When k = 0 , we have g s = { 0 } = g 0 . Assuming k = i holds, g s − i ⊆ g i , we’ll prove that for k = i + 1 , g s − i − 1 ⊆ g i + 1 . According to Equation (2), we have g s − i = [ g , g s − i − 1 ] . Also, since g s − i ⊆ g i , then g s − i − 1 ⊆ g i + 1 . Thus, for any i ( 0 ≤ i ≤ s ) such that g s − i ⊆ g i , we have g s ⊇ g 0 = g , and since g s ⊆ g , we conclude that g s = g .

Next, we prove that g 0 = 0 ⊂ g 1 ⊂ g 2 ⊂ ⋯ ⊂ g s − 1 ⊂ g s = g , namely, for any k ( 0 ≤ k ≤ s − 1 ) such that dim g k < dim g k + 1 , then g k ⊂ g k + 1 is strict.

When k = 0 , by conclusion (a), we have g 0 ⊂ g 1 . Assuming for k = i holds, g i ⊂ g i + 1 , we’ll prove that for k = i + 1 , g i + 1 ⊂ g i + 2 . According to Equation (5), we have g i + 1 = { X ∈ g | [ X , g ] ⊆ g i } , g i + 2 = { X ∈ g | [ X , g ] ⊆ g i + 1 } , thus g i + 1 ⊂ g i + 2 . Also, since g s = g , then for any k ( 0 ≤ k ≤ s − 1 ) such that dim g k < dim g k + 1 , we have g k ⊂ g k + 1 is strict. Thus, we have

g 0 = 0 ⊂ g 1 ⊂ g 2 ⊂ ⋯ ⊂ g s − 1 ⊂ g s = g

holds.

Before introducing the nilpotent complex structure on G, let’s first discuss under what conditions g is a complex Lie algebra. Let g C denote the complexification of g, and let J be the complex structure on the Lie algebra g. Then we

have g C = g − ⊕ g + , where ± i are the eigenvalues of J, and g − = { X + i J X | X ∈ g } and g + = { X − i J X | X ∈ g } are the eigenspaces of J.

Property 3 [9] . The eigenspaces g ± of J are ideals of g C .

Theorem 1. Let J be the integrable complex structure on the Lie algebra g. If J [ X , Y ] = [ J X , Y ] for all X , Y ∈ g , then g is a complex Lie algebra.

Proof: Let g + = { X − i J X | X ∈ g } . According to Property 3, we have [ X , Y ] ∈ g + for all X , Y ∈ g + .

Let g − = { X + i J X | X ∈ g } . According to Property 3, we have [ X , Y ] ∈ g − for all X , Y ∈ g − .

Since g ⊂ g C = g − ⊕ g + , any X , Y ∈ g satisfies X = X 1 + X 2 ( X 1 ∈ g − , X 2 ∈ g + ), Y = Y 1 + Y 2 ( Y 1 ∈ g − , Y 2 ∈ g + ).

According to Property 3, we have [ X 1 , Y 2 ] ∈ g + and [ X 1 , Y 2 ] ∈ g − . Since g + ∩ g − = { 0 } , it follows that [ X 1 , Y 2 ] = 0 and [ X 2 , Y 1 ] = 0 .

To prove that g is a complex Lie algebra, we need to show that the Lie bracket of g is C-linear, i.e., [ ( a + i b ) X , Y ] = ( a + i b ) [ X , Y ] ( X , Y ∈ g ).

[ ( a + i b ) X , Y ] = [ ( a + i b ) ( X 1 + X 2 ) , Y 1 + Y 2 ] = [ a ( X 1 + X 2 ) , Y 1 + Y 2 ] + [ i b ( X 1 + X 2 ) , Y 1 + Y 2 ] = a [ X 1 + X 2 , Y 1 + Y 2 ] + b [ i ( X 1 + X 2 ) , Y 1 + Y 2 ] = a [ X 1 + X 2 , Y 1 + Y 2 ] + b [ J X 1 − J X 2 , Y 1 + Y 2 ] = a [ X 1 + X 2 , Y 1 + Y 2 ] + b [ J X 1 , Y 1 + Y 2 ] − b [ J X 2 , Y 1 + Y 2 ]

= a [ X 1 + X 2 , Y 1 + Y 2 ] + b J [ X 1 , Y 1 + Y 2 ] − b J [ X 2 , Y 1 + Y 2 ] = a [ X 1 + X 2 , Y 1 + Y 2 ] + b J [ X 1 , Y 1 ] − b J [ X 2 , Y 2 ] = a [ X 1 + X 2 , Y 1 + Y 2 ] + i b [ X 1 , Y 1 ] + i b [ X 2 , Y 2 ] = ( a + i b ) [ X 1 + X 2 , Y 1 + Y 2 ]

Therefore, the Lie algebra is C-linear, proving that g is a complex Lie algebra.

Suppose G is a connected simply connected nilpotent Lie group, g = L i e G , and g has a complex structure J. Then g is a complex vector space, but generally not a complex Lie algebra. According to Theorem 1, if the complex structure satisfies J [ X , Y ] = [ J X , Y ] for all X , Y ∈ g , then g is a complex Lie algebra. To study the nilpotent complex structure of the Lie group G, we introduce the ascending sequence { a k , k ≥ 0 } related to the nilpotent Lie algebra g.

Definition 2 [10] . Let G be a connected simply connected s-step nilpotent Lie group, and suppose it has a left-invariant integrable almost complex structure. g = L i e G and J is the complex structure on the Lie algebra g. g has the sequence

a 0 = { 0 } , ⋯ , a k = { X ∈ g | [ X , g ] ⊆ a k − 1 , [ J X , g ] ⊆ a k − 1 } , ⋯ , (7)

called the compatible with the integrable almost complex structure J of G.

Property 4. If { a k , k ≥ 0 } is the compatible with the integrable almost complex structure J of G, then a_k is an ideal of g, a k ⊆ a k + 1 , and a k ⊆ g k ( k ≥ 0 ) .

Proof: We use induction to prove a k ⊆ a k + 1 .

When k = 0 , because [ a 1 , g ] ⊆ a 0 = { 0 } , then [ a 1 , g ] ⊆ a 0 ⊆ a 1 . Suppose that when k = i , a i ⊆ a i + 1 holds, we need to prove that when k = i + 1 , a i + 1 ⊆ a i + 2 . According to Equation (7), we have a i + 2 = { X ∈ g | [ X , g ] ⊆ a i + 1 , [ J X , g ] ⊆ a i + 1 } , a i + 1 = { X ∈ g | [ X , g ] ⊆ a i , [ J X , g ] ⊆ a i } . Since a i ⊆ a i + 1 holds, a i + 1 ⊆ a i + 2 , hence a k ⊆ a k + 1 ( k ≥ 0 ) .

To prove that a_k is an ideal of g, because [ a k , g ] ⊆ a k − 1 ⊆ a k , a_k is an ideal of g. By induction, we prove a k ⊆ g k ( k ≥ 0 ) .

When k = 0 , since a 0 = 0 , g 0 = 0 , then a 0 ⊆ g 0 . Suppose that when k = i , a i ⊆ g i holds, we need to prove that when k = i + 1 , a i + 1 ⊆ g i + 1 .

According to Equations (5) and (7), we have

a i + 1 = { X ∈ g | [ X , g ] ⊆ a i , [ J X , g ] ⊆ a i } , g i + 1 = { X ∈ g | [ X , g ] ⊆ g i } .

Since a i ⊆ g i holds, a i + 1 ⊆ g i + 1 , which completes the proof.

Lemma 1 [1] . If there exists k ≥ 0 such that a k = a k + 1 , then for any r ≥ k , a r = a k .

Now let’s look at some properties between the ascending sequence { a k , k ≥ 0 } and the ascending central sequence { g k , k ≥ 0 } of g.

Lemma 2 [1] . If { a k , k ≥ 0 } and { g k , k ≥ 0 } are respectively the ascending sequences of g, then the following three conclusions hold.

i) If there exists k ≥ 0 such that a k = g k , then g k = J ( g k ) ;

ii) If there exists k > 0 such that a k − 1 = g k − 1 , then a k = g k if and only if g k = J ( g k ) ;

iii) If a k − 1 = g k − 1 , then a_k is the largest J-invariant subspace of g_k.

Under the conditions of Lemma 2 and according to Equations (5), (7), we know a 0 = g 0 = 0 , then

a) If there exists k ≥ 0 such that J ( g k ) ⊄ g k , then a k ⊂ g k is strict;

b) If there exists an integer s such that a s − 1 = g s − 1 , then a s = g s = g ;

c) a₁ is the largest J-invariant subspace of g₁;

d) If any term g_k of { g k , k ≥ 0 } is J-invariant, then for any k ≥ 0 , a k = g k and a s = g s = g .

Next, consider some related properties between the ascending sequence { a k , k ≥ 0 } and the descending central sequence { g l , l ≥ 0 } of g. Under the conditions of Lemma 2, suppose { g l , l ≥ 0 } is the descending central sequence of g. Then we have

① If for some k ≥ 0 and some l ≥ 0 , g l ⊂ a k and J ( g l − 1 ) = g l − 1 , then g l − 1 ⊂ a k + 1 ;

② If for some k ≥ 0 , [ g , g ] ⊆ a k , then a k + 1 = g ;

③ If any g l ∈ { g l , l ≥ 0 } is J-invariant, then a s = g s = g [1] .

Definition 3 [11] . Let G be a connected simply connected s-step nilpotent Lie group, and suppose it has a left-invariant integrable almost complex structure. g = L i e G and J is the complex structure on the Lie algebra g. If there exists t > 0 such that a t = g , then the left-invariant integrable almost complex structure J is called a nilpotent left-invariant complex structure.

Lemma 3 [1] . Let { w i , 1 ≤ i ≤ n } be a (1,0)-type left-invariant form complex basis for g * , satisfying the structural equation

∑ j < k < i A i j k w j ∧ w k + ∑ j , k < i B i j k w j ∧ w ¯ k ( 1 ≤ i ≤ n ).

If { Z i , Z ¯ i , 1 ≤ i ≤ n } is a basis for g and dual to the basis { w i , w ¯ i , 1 ≤ i ≤ n } , let X i = Re ( Z i ) , Y i = Im ( Z i ) ( 1 ≤ i ≤ n ) , then any term a l ( 1 ≤ l ≤ n ) in the ascending sequence { a l , l ≥ 0 } contains at least generators X n − l + 1 , Y n − l + 1 , ⋯ , X n , Y n .

Proposition 1. Under the conditions of Lemm3, we have

(i) If a_k is a member of the sequence { a k , k ≥ 0 } , and a k ≠ g , then dim a k + 1 ≥ 2 + dim a k ;

(ii) If { g k , k ≥ 0 } is the ascending central sequence of g, then dim g k ≥ dim a k ≥ 2 k ( 1 ≤ k ≤ n ) ;

(iii) There exists a unique integer t such that dim a t − 1 < dim a t and a t = g ( s ≤ t ≤ n ) .

Theorem 2 [1] . Let { a k , k ≥ 0 } be the compatible with the integrable almost complex structure J of G. If there exists a(1,0)-type left-invariant form complex basis { w i , 1 ≤ i ≤ n } for g * such that the basis satisfies the structural equation

d w i = ∑ j < k < i A i j k w j ∧ w k + ∑ j , k < i B i j k w j ∧ w ¯ k ( 1 ≤ i ≤ n ),

then the left-invariant integrable almost complex structure is nilpotent if and only if it is almost nilpotent.

Raghunathan [12] concludes: let G be a connected simply connected nilpotent Lie group with Lie algebra g = L i e G . G has a lattice D if and only if g admits a basis with rational structure constants. By applying Theorem 2, Theorem 3 can be obtained.

Theorem 3 [1] . Given the structure equations of Theorem 2:

d w i = ∑ j < k < i A i j k w j ∧ w k + ∑ j , k < i B i j k w j ∧ w ¯ k ( 1 ≤ i ≤ n ),

a connected simply connected nilpotent Lie group G with a left-invariant almost complex structure that is nilpotent can be defined, and its left-invariant complex structure is nilpotent. Then a complex structure, which is also nilpotent, can be defined on the compact homogeneous nilpotent manifold G/D. Conversely, if the left-invariant integrable almost complex structure of a connected simply connected nilpotent Lie group G is nilpotent, with structure equations

d w i = ∑ j < k < i A i j k w j ∧ w k + ∑ j , k < i B i j k w j ∧ w ¯ k ( 1 ≤ i ≤ n ),

then G has a left-invariant complex structure that is nilpotent.

Definition 4. Let G be a connected simply connected nilpotent Lie group with a left-invariant integrable almost complex structure, g = L i e G , and J be the complex structure on g. Define the mapping on g / a k :

J k : g / a k → g / a k

x ¯ ↦ J ˜ ( x ¯ ) = J ( x ) + a k , x ∈ g , x ¯ ∈ g / a k .

Lemme 5. Suppose G is a connected simply connected nilpotent Lie group with a left-invariant integrable almost complex structure that is nilpotent, then G_k has a left-invariant integrable almost complex structure.

Proof: First, we prove that J_k is a complex structure. Since J is linear, and according to Definition 4,

J k ( x ¯ ) = J ( x ) + a k , x ∈ g , x ¯ ∈ g / a k ,

J_k is linear. Next, we prove that J_k is a complex structure. Since

J k ( J k ( x ¯ ) ) = J k ( J x + a k ) = J ( J x + a k ) + a k = J 2 x + a k = − x ¯

thus J k 2 = − i d : g / a k → g / a k , so J_k is a complex structure on g / a k .

For any X ¯ , Y ¯ ∈ g / a k , we have X ¯ = X + a k , Y ¯ = Y + a k ( X , Y ∈ g ),

[ J k ( X ¯ ) , J k ( Y ¯ ) ] = [ J X + a k , J Y + a k ] = [ J X , J Y ] + a k = [ X ¯ , Y ¯ ] + J [ J X , Y ] + J [ X , J Y ] + a k = [ X ¯ , Y ¯ ] + J k ( [ J X , Y ] + a k ) + J k ( [ X , J Y ] + a k ) = [ X ¯ , Y ¯ ] + J k ( [ J k ( X ¯ ) , Y ¯ ] ) + J k ( [ X ¯ , J k ( Y ¯ ) ] )

so G_k has a left-invariant integrable almost complex structure. □

We know that the left-invariant integrable almost complex structure J of the Lie group G induces a left-invariant integrable almost complex structure J_k on the Lie group G_k. Next, we first give a sequence { b k , k ≥ 0 } on g / a k , and then use this sequence to prove that if J is nilpotent, then J_k is also nilpotent.

Definition 5. Let G be a connected simply connected nilpotent Lie group with a left-invariant integrable almost complex structure, g = L i e G , J be the complex structure on g, and J_k be the complex structure on g / a k . g / a k has a sequence

b 0 = { 0 } , ⋯ , b k = { X ¯ ∈ g / a k | [ X ¯ , g / a k ] ⊆ b k − 1 , [ J ˜ ( X ¯ ) , g / a k ] ⊆ b k − 1 } , ⋯ . (8)

Property 5. g / a k has a sequence { b k , k ≥ 0 } satisfying Equation (8), which implies that

(1) For any k ≥ 0 , b k ⊆ b k + 1 ;

(2) If there exists k ≥ 0 such that b k = b k + 1 , then for any r ≥ k , b r = b k .

Proof: We use induction to prove b k ⊆ b k + 1 .

When k = 0 , according to Equation (8), we have b 0 = { 0 } and b 0 ⊆ b 1 .

Assume that when k = i , b i ⊆ b i + 1 holds. When k = i + 1 , b i + 1 ⊆ b i + 2 holds. According to Equation(8), we have

b i + 2 = { X ¯ ∈ g / a k | [ X ¯ , g / a k ] ⊆ b i + 1 , [ J k ( X ¯ ) , g / a k ] ⊆ b i + 1 } ,

b i + 1 = { X ¯ ∈ g / a k | [ X ¯ , g / a k ] ⊆ b i , [ J k ( X ¯ ) , g / a k ] ⊆ b i } ,

and since b i ⊆ b i + 1 holds, b i + 1 ⊆ b i + 2 holds. Thus, b k ⊆ b k + 1 ( k ≥ 0 ).

Next, we prove property (2). According to Equation (8), we have

b i + 2 = { X ¯ ∈ g / a k | [ X ¯ , g / a k ] ⊆ b i + 1 , [ J k ( X ¯ ) , g / a k ] ⊆ b i + 1 } = { X ¯ ∈ g / a k | [ X ¯ , g / a k ] ⊆ b i , [ J k ( X ¯ ) , g / a k ] ⊆ b i } = b k + 1

which means that for any r ≥ k , b r = b k + 1 = b k . □

Definition 6. Let G be a connected simply connected nilpotent Lie group with a left-invariant integrable almost complex structure, g = L i e G , J_k be the complex structure on g / a k , and { b k , k ≥ 0 } be an ascending sequence on g / a k . If there exists t > 0 such that b t = g / a k , then the left-invariant integrable almost complex structure of G_k is called nilpotent left-invariant complex structure.

Theorem 4. Suppose a connected simply connected s-step real nilpotent Lie group G has a left-invariant integrable almost complex structure J, and the sequence { a k , k ≥ 0 } is a ascending sequence of g. If the left-invariant integrable almost complex structure J is nilpotent, then b n = a n + k / a k and the left-invariant integrable almost complex structure of G_k is nilpotent.

Proof: To prove that the left-invariant integrable almost complex structure of G_k is nilpotent, we only need to prove that there exists t such that b t = g / a k . We will prove by induction that there exists n such that b n = g / a k .

b 0 = a k / a k = { 0 } , Next, we prove b 1 = a k + 1 / a k .

For any X ¯ ∈ a k + 1 / a k , we have X ¯ = X + a k ( X ∈ a k + 1 ) , then [ X ¯ , g / a k ] = [ X , g ] + a k and [ J k ( X ¯ ) , g / a k ] = [ J X , g ] + a k According to Equation (7), we know that [ X , g ] ⊆ a k and [ J ( X ) , g ] ⊆ a k , so

[ J k ( X ¯ ) , g / a k ] = 0 and [ X ¯ , g / a k ] = 0 .

Therefore X ¯ ∈ b 1 , implying b 1 ⊇ a k + 1 / a k .

For any X ¯ ∈ b 1 , we have X ¯ = X + a k ( X ∈ g ) , according to Equation (8),

b 1 = { X ¯ ∈ g / a k | [ X ¯ , g / a k ] ⊆ b 0 = { 0 } , [ J k ( X ¯ ) , g / a k ] ⊆ b 0 = { 0 } } ,

which means [ X , g ] + a k ⊆ { 0 } and [ J X , g ] + a k ⊆ { 0 } , so [ X , g ] ⊆ a k and [ J X , g ] ⊆ a k , hence X ∈ a k + 1 , so X ¯ ∈ a k + 1 / a k , which means b 1 ⊆ a k + 1 / a k .

So b 1 = a k + 1 / a k .

Assume that when n = i , b i = a k + i / a k holds.

Next, we prove that when n = i + 1 , b i + 1 = a k + i + 1 / a k holds.

For any X ¯ ∈ b i + 1 we have X ¯ = X + a k ( X ∈ a k + i + 1 ) , then [ X ¯ , g / a k ] = [ X , g ] + a k and [ J k ( X ¯ ) , g / a k ] = [ J X , g ] + a k . according to Equation (8), we know that [ X , g ] ⊆ a k + i and [ J X , g ] ⊆ a k + i . thus [ J k ( X ¯ ) , g / a k ] ⊆ a k + i / a k = b i and [ X ¯ , g / a k ] ⊆ a k + i / a k = b i . Therefore X ¯ ∈ b i + 1 , implying b i + 1 ⊇ a k + i + 1 / a k .

For any X ¯ ∈ b i + 1 , we have X ¯ = X + a k ( X ∈ g ) , according to Equation (8), b i + 1 = { X ¯ ∈ g / a k | [ X ¯ , g / a k ] ⊆ b i = a k + i / a k , [ J k ( X ¯ ) , g / a k ] ⊆ b i = a k + i / a k } , which means [ X , g ] + a k ⊆ a k + i / a k and [ J X , g ] + a k ⊆ a k + i / a k , so [ X , g ] ⊆ a k + i and [ J X , g ] ⊆ a k + i , hence X ∈ a k + i + 1 , so X ¯ ∈ a k + i + 1 / a k , which means b i + 1 ⊆ a k + i + 1 / a k .

So b i + 1 = a k + i + 1 / a k .

Next, we prove that the complex structure J_k on g/a_k is nilpotent. Since the complex structure J on the Lie group G is nilpotent, there exists t > 0 such that a t = g , then b t − k = a t / a k = g / a k , thus J_k is nilpotent.

According to Theorem4, if a connected simply connected s-step real nilpotent Lie group G has a left-invariant integrable almost complex structure J that is nilpotent, then J can induce a nilpotent left-invariant integrable almost complex structure J_k on the Lie group G_k.