lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -303,16 +303,16 @@ Z \in \mathfrak{gl}_n(K)\). In fact this same identity show\dots
 \end{lemma}
 
 The reason why we are disccussing invariant bilinear forms is the following
-characterization of finite-dimensional semisimple Lie algebras.
+characterization of finite-dimensional semisimple Lie algebras, known as
+\emph{Cartan's criterium for semisimplicity}.
 
-% TODO: Prove this
 \begin{proposition}
   Let \(\mathfrak{g}\) be a Lie algebra. The following statements are
   equivalent.
   \begin{enumerate}
     \item \(\mathfrak{g}\) is semisimple.
-    \item For each finite-dimensional representation \(V\) of \(\mathfrak{g}\),
-      the \(\mathfrak{g}\)-invariant bilinear form
+    \item For each non-trivial finite-dimensional representation \(V\) of
+      \(\mathfrak{g}\), the \(\mathfrak{g}\)-invariant bilinear form
       \begin{align*}
         B_V : \mathfrak{g} \times \mathfrak{g} & \to K \\
         (X, Y) &
@@ -325,8 +325,23 @@ characterization of finite-dimensional semisimple Lie algebras.
   \end{enumerate}
 \end{proposition}
 
-We refer the reader for \cite[ch. 5]{humphreys} for a proof of this last
-result. Without further ado, we may proceed to our\dots
+This proof is somewhat techinical, but the idea behind it is simple. First, for
+\strong{(i)} \(\implies\) \strong{(ii)} we show that \(\mathfrak{a} = \{ X \in
+\mathfrak{g} : B_V(X, Y) = 0 \, \forall Y \in \mathfrak{g}\}\) is a solvable
+ideal of \(\mathfrak{g}\). Hence \(\mathfrak{a} = 0\). For \strong{(ii)}
+\(\implies\) \strong{(iii)} it suffices to take \(V = \mathfrak{g}\) the
+adjoint representation. Finally, for \strong{(iii)} \(\implies\) \strong{(i)}
+we note that the orthogonal complement of any \(\mathfrak{a} \normal
+\mathfrak{g}\) with respect to the Killing form \(B\) is an ideal
+\(\mathfrak{b}\) of \(\mathfrak{g}\) with \(\mathfrak{g} = \mathfrak{a} \oplus
+\mathfrak{b}\). Furtheremore, the Killing form of \(\mathfrak{a}\) is the
+restriction \(B\!\restriction_{\mathfrak{a}}\) of the Killing form of
+\(\mathfrak{g}\) to \(\mathfrak{a} \times \mathfrak{a}\), which is
+non-degenerate. It then follows from induction in \(\dim \mathfrak{a}\) that
+\(\mathfrak{g}\) is the sum of simple ideals.
+
+We refer the reader to \cite[ch. 5]{humphreys} for a complete proof. Without
+further ado, we may proceed to our\dots
 
 \section{Proof of Complete Reducibility}