Is the set of nilpotents of a Lie algebra a Lie subalgebra?

Question

Let $\mathfrak{g}\subset gl(V)$ be a Lie algebra. Is the set of nilpotents of $\mathfrak{g}$ a lie subalgebra? To be more precise, let $A$ and $B$ be nilpotent matrices. Then is $AB-BA$ also nilpotent?

Answer 1

No. Take $A=\pmatrix{0&1\\0&0}$ and $B=\pmatrix{0&0\\1&0}$.