Let $\frak g$ be a complex semisimple Lie algebra. Is every $X\in\frak g$ contained in some Borel subalgebra $\frak b$?
Attempt: I know that a Borel subalgebra is by definition a maximal solvable subalgebra. Now, $Span(X)$ is abelian so it is solvable. Hence $Span(X)$ is contained in a maximal solvable subalgebra.
I am not sure about the last part. Can we justify this?