Why is the realification of a simple complex Lie algebra a semisimple real Lie algebra?

Question

Why is the realification of a simple complex Lie algebra a semisimple real Lie algebra?

The realification here means to consider the complex Lie algebra as a real Lie algebra of twice the dimension.

The statement was used in the proof of Proposition 12.46 in https://doi.org/10.1016/S0079-8169(08)61672-4.

Answer 1

By the Cartan criterion, a Lie algebra $\mathfrak g$ is semisimple if and only of its Killing form is non-degenerate. So, if $\mathfrak g$ is a semisimple complex Lie algebra, its Kiling form is non-degenerate, but that Killing form is also the Killing form of the realification of $\mathfrak g$.

Answer 2

Actually, a stronger statement holds true: Denoting the "realification" of a complex Lie algebra $L$ by $Res_{\mathbb C\vert \mathbb R}L$, then

$L$ is simple iff $Res_{\mathbb C\vert \mathbb R}L$ is simple.

Namely, for the non-trivial direction, say we've already shown that $Res_{\mathbb C\vert \mathbb R}L$ is semisimple like in José Carlos Santos' answer. Let $S$ be a simple component of $Res_{\mathbb C\vert \mathbb R}L$. Now for any $0 \neq \lambda \in \mathbb C$, we have that $\lambda S$ is also an ideal of $Res_{\mathbb C\vert \mathbb R}L$, and $[S, \lambda S] = \lambda S \neq 0$ hence by simplicity $S \subseteq \lambda S$. But then we actually have equality $S=\lambda S$, because they have the same real dimension. But that means the complex span of $S$ in $L$ is $S$ itself, hence it is a non-zero ideal of $L$, hence all of $L$ by $L$ being simple, and hence as a set it is also all of $Res_{\mathbb C\vert \mathbb R}L$.

This statement and proof, including the step provided by José Carlos Santos' answer, works far more generally for any finite extension of characteristic $0$ fields $K\vert k$ instead of $\mathbb C \vert \mathbb R$, and I took the proof from Bourbaki's Lie Groups and Algebras, chapter I §6 no. 10.

If one digs deeper into the theory, actually the following holds true: If the Dynkin diagram to $L$ is of type $R$ (i.e. $A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4$ or $G_2$), then $Res_{\mathbb C\vert \mathbb R}L$ is a quasi-split real Lie algebra whose Satake diagram consists of two copies of the Dynkin diagram of $L$, with complex conjugation flipping those two copies. That real Lie algebra is simple, but not "absolutely simple", since if one complexifies it again, one gets a direct sum of two copies of the original $L$. Again, this holds true in greater generality, compare section 4.1 (especially p. 67) of my thesis.