lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -1816,14 +1816,16 @@ the primary tool we'll use to generalize the results of the previous section.
 What is simultaneous diagonalization all about then?
 
 \begin{definition}\label{def:sim-diag}
-  A set of square matrices \(\{X_i\}_i\) is called \emph{simultaneously
-  diagonalizable} if there some invertible matrix \(P\) such that \(P X_i
-  P^{-1}\) is diagonal for every \(i\).
+  Given a \(K\)-vector space \(V\), a set of operators \(\{T_j : V \to V\}_j\)
+  is called \emph{simultaneously diagonalizable} if there is a basis \(\{v_1,
+  \ldots, v_n\}\) for \(V\) such that \(T_j v_i\) is a scalar multiple of
+  \(v_i\), for all \(i, j\).
 \end{definition}
 
 \begin{proposition}
-  A set of diagonalible matrices is simultaneously diagonalizable if, and only
-  if all of its elements commute with one another.
+  Given a \emph{finite-dimensional} vector space \(V\), A set of diagonalizable
+  operators \(V \to V\) is simultaneously diagonalizable if, and only if all of
+  its elements commute with one another.
 \end{proposition}
 
 \begin{corollary}
@@ -1843,19 +1845,7 @@ What is simultaneous diagonalization all about then?
 % TODOO: Prove that the operators are diagonalizable
 \begin{proof}
   It suffices to show that \(H : V \to V\) is a diagonalizable operator for
-  each \(H \in \mathfrak{h}\). Indeed, if this is the case it follows from the
-  fact that \(\mathfrak{h}\) is Abelian that the set
-  \(\{[H\!\restriction_V]_{\mathcal{B}} : H \in \mathfrak{h}\}\) is
-  simultaneously diagonalible for any basis \(\mathcal{B}\) for \(V\).
-
-  If \(P\) is as in definition~\ref{def:sim-diag} and \(T\) is the operator \(V
-  \to V\) such that \([T]_{\mathcal{B}}\), then \(\mathcal{C} =
-  T(\mathcal{B})\) is a basis for \(V\) such that
-  \([H\!\restriction_V]_{\mathcal{C}} = P [H\!\restriction_V]_{\mathcal{B}}
-  P^{-1}\) is diagonal for each \(H \in \mathfrak{h}\). If we then take
-  \(\{v_1, \ldots, v_n\} = \mathcal{C}\) and denote by \(\lambda_i(H)\) the
-  coefficient of \(E_{i i}\) in \([H\!\restriction_V]_{\mathcal{C}}\), we find
-  \(H v_i = \lambda_i(H) \cdot v_i\) as intended.
+  each \(H \in \mathfrak{h}\). 
 \end{proof}
 
 As promised, this implies\dots