Is the direct sum of the strictly upper triangular matrices and the lower triangular matrices = the general linear algebra?

Question

I'm just trying to make sure I understand direct sum correctly in the context of Lie algebra, this isn't a textbook question.

If we, for example, took the strictly upper triangular matrices in the complex plane n(n,C) and performed a direct sum with the lower triangular matrices l(n,C) would that yield the general linear Lie algebra?

Answer 1

As vector spaces we have indeed $\mathfrak{gl}_n(K)=\mathfrak{n}_n(K)\oplus \mathfrak{l}_n(K)$, but not as Lie algebras, because these subalgebras are not ideals in $\mathfrak{gl}_n(K)$. In fact, the subalgebra of (strictly) upper triangular matrices is not an ideal for $n\ge 2$.