lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -430,7 +430,7 @@ we introduce a distinguished element of \(\mathcal{U}(\mathfrak{g})\), known as
   all representations \(V\) of \(\mathfrak{g}\): its action commutes with the
   action of any other element of \(\mathfrak{g}\).
 
-  % TODOOO: Prove that the action is not zero when V is non-trivial
+  % TODOO: Prove that the action is not zero when V is non-trivial
   In particular, it follows from Schur's lemma that if \(V\) is
   finite-dimensional and irreducible then \(C\) acts in \(V\) as a scalar
   operator.
@@ -1799,14 +1799,14 @@ simple calculation also shows that if \(i \ne j\) then the coefficient of
 in \(X\), for all \(X \in \mathfrak{gl}_n(K)\). In particular, if \([E_{i i},
 X]\) is diagonal for all \(i\), then so is \(X\) -- i.e. \(\mathfrak{h}\) is
 self-normalizing. Hence \(\mathfrak{h}\) is a Cartan subalgebra of
-\(\mathfrak{gl}_n(K)\). The intersection of such subalgebra with
-\(\mathfrak{sl}_n(K)\) -- i.e. the subalgebra of traceless diagonal matrices --
-is a Cartan subalgebra of \(\mathfrak{sl}_n(K)\). In particular, if \(n = 2\)
-or \(n = 3\) we get to the subalgebras described the previous two sections.
-
-The remaining question then is: if \(\mathfrak{h} \subset \mathfrak{g}\) is a
-Cartan subalgebra and \(V\) is a representation of \(\mathfrak{g}\), does the
-eigenspace decomposition
+\(\mathfrak{gl}_n(K)\).
+
+The intersection of such subalgebra with \(\mathfrak{sl}_n(K)\) -- i.e. the
+subalgebra of traceless diagonal matrices -- is a Cartan subalgebra of
+\(\mathfrak{sl}_n(K)\). In particular, if \(n = 2\) or \(n = 3\) we get to the
+subalgebras described the previous two sections. The remaining question then
+is: if \(\mathfrak{h} \subset \mathfrak{g}\) is a Cartan subalgebra and \(V\)
+is a representation of \(\mathfrak{g}\), does the eigenspace decomposition
 \[
   V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda
 \]