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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?

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Every solvable subalgebra $S$ of $L$ is contained in some Borel subalgebra $B$ of $L$. This is the definition of a Borel subalgebra. Over the complex numbers all Borel subalgebras are conjugated.

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