Given a subspace V of a metric Lie algebra, does [[V,V],V]⊂V imply [V,V]⊂V?

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Cleveland Walters · Accepted Answer

Step 1  Take  g=R3  with  [x,y]=x×y  the cross product and as symmetric bilinear form ⟨⋅,⋅⟩ the usual scalar product.  Then the space  V=⟨e1,e2⟩  is not closed under the Lie bracket since  [e1,e2]=e1×e2=e3  However, since we have  [V,V]=⟨e3⟩  we find that  [[V,V],V]⊂qV  even though V is not a Lie subalgebra.

Mary Goodson · Accepted Answer

Step 1
My answer to another of your questions also applies here. Take a Cartan decomposition of a semisimple Lie algebra or more generally a symmetric decomposition.
A symmetric decomposition is defined as

g = h \oplus m

where

[h, h] \leq h, [h, m] \leq m

and

[m, m] \leq h

.
It is a quick check to see that a Cartan decomposition is symmetric. More generally these correspond to symmetric homogeneous space so you can find a whole load of examples of these. MSL Immediately you get

[m, [m, m]] \leq m

and this breaks your rule.

Given a subspace V of a metric Lie algebra, does

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