lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -305,6 +305,17 @@ is only natural to define\dots
 \end{example}
 
 \begin{example}
+  Let \(\mathfrak{g}_1\) and \(\mathfrak{g}_2\) be a Lie algebras over \(K\).
+  Then the space \(\mathfrak{g}_1 \oplus \mathfrak{g}_2\) is a Lie algebra with
+  brackets
+  \[
+    [X_1 + X_2, Y_1 + Y_2] = [X_1, Y_1] + [X_2, Y_2],
+  \]
+  and \(\mathfrak{g}_1, \mathfrak{g}_2 \normal \mathfrak{g}_1 \oplus
+  \mathfrak{g}_2\).
+\end{example}
+
+\begin{example}
   Let \(G\) be an affine algebraic \(K\)-group and \(H \subset G\) be a
   connected closed subgroup. Denote by \(\mathfrak{g}\) and \(\mathfrak{h}\)
   the Lie algebras of \(G\) and \(H\), respectively. The inclusion \(H \to G\)
@@ -1022,7 +1033,6 @@ This last proposition is known as \emph{Frobenius reciprocity}, and was first
 proved by Frobenius himself in the context of finite-groups. Another
 interesting construction is\dots
 
-% TODO: Define the direct sum of Lie algebras!
 \begin{example}
   Let \(\mathfrak{g}\) and \(\mathfrak{h}\) be Lie algebras. Given a
   \(\mathfrak{g}\)-module \(V\) and a \(\mathfrak{h}\)-module \(W\), the space