= Solution
Take the <Lie subalgebra> $\mathfrak b\subseteq\mathfrak{gl}_2(\mathbb C)$ of all upper-triangular matrices. It is a <solvable Lie algebra> with a nonzero abelian <ideal of a Lie algebra>, hence is not a <semisimple Lie algebra>. For every $X\in\mathfrak b$, the polynomial description $X_s=p(X)$ in the <Additive Jordan decomposition> keeps $X_s$ upper triangular; then $X_n=X-X_s$ is upper triangular too. Thus \b[the upper-triangular matrix algebra is closed under <Additive Jordan decomposition>] despite being nonsemisimple. A diagonal matrix algebra gives an even simpler <abelian Lie algebra> example.
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