lie-algebras-and-their-representations

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