I'm studying Varadarajan's Lie algebra, and I think one of the proofs uses this fact without explicitly proving it:
If $\phi:g\rightarrow gl(V)$ is a finite dimensional representation of an arbitrary finite dimensional Lie algebra over a characteristic zero field $k$, if $k'$ is the algebraic closure of $k$, then $\phi'$ may be extended to a representation over $k'$. Is it true that $\phi$ is semisimple iff $\phi'$ is semisimple? (One direction is obvious. I'm confused about the other direction)
My field theory is weak, any help would be appreciated!
(I'm aware of this post Complete reducibility of a field extension of an lie algebra representation but this post it's actually something different.)