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}