vector spaces: sum of the maximum abelian ideal and attached map cores in a nilpotent Lie algebra

Leave $ mathbb {K} $ be a field with $ char ( mathbb {K}) neq 2 $ Y $ L $ a nilpotent lie algebra of finite dimension on $ mathbb {K} $.

Suppose that $ dim L = 5 + dim Z (L) $ Y $ L = span {x, y, z, u, v } oplus Z (L) $ where $ dim A = 2 + dim Z (L) $ Y $ A = span {u, v } oplus Z (L) $ It is the ultimate abelian ideal.

Define $ H_ {a + x}: = A + ker (ad_ {a + x}) $ for all $ a in A $. So $ x + A in H_ {a + x} / A $ for all $ a en A setminus Z (L) $. Suppose we have $ dim (H_x / A) = dim (H_ {a + x} / A) = 2 $ for all $ a en A setminus Z (L) $.

Can we show that it exists $ a neq b en A setminus Z (L) $ such that $ H_x subseteq H_ {a + x} $ or $ H_ {a + x} subseteq H_x $ or $ H_ {a + x} subseteq H_ {b + x} $?

Now we can prove the statement when $ mathbb {K} = mathbb {F} _q $ It is a finite field as follows. Note that $ L / A = span {x + A, y + A, z + A } $ Y $ H_x / A = span {x + A, alpha_0y + beta_0z + A } $ for some $ (0, 0) neq ( alpha_0, beta_0) in mathbb {F} _q ^ 2 $ and for each $ a en A setminus Z (L), H_ {a + x} / A = span {x + A, alpha_ay + beta_az + A } $ for some $ (0, 0) neq ( alpha_a, beta_a) in mathbb {F} _q ^ 2 $. As $ dim (A / Z (L)) = 2 $, $ | {H_ {a + x} | a en A setminus Z (L) } cup {H_x } | leq q ^ 2 $. Keep in mind that there is at most $ q ^ 2 – 1 $ form subspaces $ span {x + A, alpha and + beta z + A } $ where $ (0, 0) neq ( alpha, beta) in mathbb {F} _q ^ 2 $. Therefore, we have the result of the fourth theorem of isomorphism.

Is it still true when $ mathbb {K} $ It's infinite?

In case we need to use them, we can also assume that

  1. $ H_ {a + x} cap H_ {b + x} neq A $ for all $ a, b in A $
  2. $ (A, L) subseteq Z (L) $
  3. $ (L, L) = (A, L) $.

But I feel we don't need these conditions.