= Complexification preserves semisimplicity
{title2=$\operatorname{rad}(\mathfrak g_{\mathbb C})=(\operatorname{rad}\mathfrak g)_{\mathbb C}$}
For a finite-dimensional real <Lie algebra> $\mathfrak g$, the <solvable radical> of its <complexification of a Lie algebra> is invariant under complex conjugation. Its real fixed subspace is a solvable ideal of $\mathfrak g$, and complexifying that fixed subspace recovers the whole radical. Conversely the complexification of any real solvable ideal is solvable. These inclusions prove the displayed identity; in particular $\mathfrak g$ is semisimple if and only if $\mathfrak g_{\mathbb C}$ is semisimple.
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