diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -158,14 +158,14 @@ result. Without further ado, we may proceed to a proof of\dots
\section{Complete Reducibility}
Historically, complete reducibility was first proved by Herman Weyl for \(K =
-\mathbb{C}\), using his knowledge of unitary representations of compact groups.
-Namely, Weyl showed that any finite-dimensional semisimple complex Lie algebra
-is (isomorphic to) the complexification of the Lie algebra of a unique simply
-connected compact Lie group, known as its \emph{compact form}. Hence the
+\mathbb{C}\), using his knowledge of smooth representations of compact Lie
+groups. Namely, Weyl showed that any finite-dimensional semisimple complex Lie
+algebra is (isomorphic to) the complexification of the Lie algebra of a unique
+simply connected compact Lie group, known as its \emph{compact form}. Hence the
category of the finite-dimensional representations of a given complex
semisimple algebra is equivalent to that of the finite-dimensional smooth
representations of its compact form, whose representations are known to be
-completely reducible.
+completely reducible -- see \cite[ch. 3]{serganova} for instance.
This proof, however, is heavily reliant on the geometric structure of
\(\mathbb{C}\). In other words, there is no hope for generalizing this for some