I read in Weibel's homological algebra that Whitehead's first and second lemmas are true for any characteristic 0 field. I mean the following:
Whitehead Lemma(s): Let g be a semisimple Lie algebra over a characteristic 0 field, V any finite-dimensional representation of g, then H^1(g, V) = H^2(g, V) = 0.
The proof uses
Theorem 7.8.9 : H^i(g, M) = 0 for any nontrivial irreducible representation M of g if g is semisimple over char 0 field.
The proof of this uses the Casimir operator. The Casimir operator is said to act by a scalar, but doesn't this assertion use Schur's lemma? Schur's lemma requires the field to be algebraically closed.
Whitehead's first lemma implies the complete reducibility of finite-dimensional g-modules, but isn't complete reducibility only true for semisimple Lie algebras over an algebraically closed field of char 0?
Whitehead's second lemma implies the Levi decomposition. Does Levi decomposition require the field to be algebraically closed?
Does anyone know where I can find a proof of the Whitehead Lemma(s) (assuming it is true) when the field is not assumed to be algebraically closed?