lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -139,7 +139,7 @@ this last construction.
   \]
 \end{example}
 
-\begin{example}
+\begin{example}\label{ex:sl2-basis}
   The elements
   \begin{align*}
     e & = \begin{pmatrix} 0 & 1 \\ 0 &  0 \end{pmatrix} &
@@ -399,18 +399,20 @@ Other interesting classes of Lie algebras are the so called \emph{simple} and
 
 \begin{example}
   The Lie algebra \(\mathfrak{sl}_2(K)\). To see this, notice that any ideal
-  \(\mathfrak{a} \normal \mathfrak{sl}_2(K)\) must be stable under the adjoint
-  action of \(h\). But the operator \(\operatorname{ad}(h)\) is diagonalizable,
-  with eigenvalues \(0\) and \(\pm 2\). Hence \(\mathfrak{a}\) must be spanned
-  by some of the eigenvectors \(e, f, h\) of \(\operatorname{ad}(h)\). If \(h
-  \in \mathfrak{a}\), then \([e, h] = - 2 e \in \mathfrak{a}\) and \([f, h] = 2
-  f \in \mathfrak{a}\), so \(\mathfrak{a} = \mathfrak{sl}_2(K)\). If \(e \in
-  \mathfrak{a}\) then \([f, e] = - h \in \mathfrak{a}\), so again
-  \(\mathfrak{a} = \mathfrak{sl}_2(K)\). Similarly, if \(f \in \mathfrak{a}\)
-  then \([e, f] = h \in \mathfrak{a}\) and \(\mathfrak{a} =
-  \mathfrak{sl}_2(K)\). More generally, the Lie algebra \(\mathfrak{sl}_n(K)\)
-  is simple for each \(n > 0\) -- see the section of \cite[ch. 6]{kirillov} on
-  invariant bilinear forms and the semisimplicity of classical Lie algebras.
+  \(\mathfrak{a} \normal \mathfrak{sl}_2(K)\) must be stable under the operator
+  \(\operatorname{ad}(h) : \mathfrak{sl}_2(K) \to \mathfrak{sl}_2(K)\) given by
+  \(\operatorname{ad}(h) X = [h, X]\). But example~\ref{ex:sl2-basis} implies
+  \(\operatorname{ad}(h)\) is diagonalizable, with eigenvalues \(0\) and \(\pm
+  2\). Hence \(\mathfrak{a}\) must be spanned by some of the eigenvectors \(e,
+  f, h\) of \(\operatorname{ad}(h)\). If \(h \in \mathfrak{a}\), then \([e, h]
+  = - 2 e \in \mathfrak{a}\) and \([f, h] = 2 f \in \mathfrak{a}\), so
+  \(\mathfrak{a} = \mathfrak{sl}_2(K)\). If \(e \in \mathfrak{a}\) then \([f,
+  e] = - h \in \mathfrak{a}\), so again \(\mathfrak{a} = \mathfrak{sl}_2(K)\).
+  Similarly, if \(f \in \mathfrak{a}\) then \([e, f] = h \in \mathfrak{a}\) and
+  \(\mathfrak{a} = \mathfrak{sl}_2(K)\). More generally, the Lie algebra
+  \(\mathfrak{sl}_n(K)\) is simple for each \(n > 0\) -- see the section of
+  \cite[ch. 6]{kirillov} on invariant bilinear forms and the semisimplicity of
+  classical Lie algebras.
 \end{example}
 
 \begin{definition}\label{thm:sesimple-algebra}