lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -369,15 +369,15 @@ There is also a natural analogue of quotients.
   any linear map \(\mathfrak{g} \to \mathfrak{h}\) between Abelian Lie algebras
   \(\mathfrak{g}\) and \(\mathfrak{h}\) is a homomorphism of Lie algebras. In
   particular, any linear isomorphism \(\mathfrak{g} \isoto K^n\) -- where
-  \(K^n\) is endowed with the trivial bracket \([v, w] = 0 \, \forall v, w \in
-  K^n\) -- is an isomorphism of Lie algebras for Abelian \(\mathfrak{g}\).
+  \(K^n\) is endowed with the trivial bracket \([v, w] = 0\), \(v, w \in K^n\)
+  -- is an isomorphism of Lie algebras for Abelian \(\mathfrak{g}\).
 \end{note}
 
 \begin{example}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{z} = \{ X \in
-  \mathfrak{g} : [X, Y] = 0 \; \forall Y \in \mathfrak{g}\}\). Then
-  \(\mathfrak{z}\) is an Abelian ideal of \(\mathfrak{g}\), known as \emph{the
-  center of \(\mathfrak{z}\)}.
+  \mathfrak{g} : [X, Y] = 0, Y \in \mathfrak{g}\}\). Then \(\mathfrak{z}\) is
+  an Abelian ideal of \(\mathfrak{g}\), known as \emph{the center of
+  \(\mathfrak{z}\)}.
 \end{example}
 
 Due to their relationship with Lie groups and algebraic groups, Lie algebras