diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -614,8 +614,8 @@ obstructions to complete reducibility. Explicitly\dots
\begin{theorem}
Given \(\mathfrak{g}\)-modules \(N\) and \(L\), there is a one-to-one
- correspondence between elements of \(H^1(\mathfrak{g}, \operatorname{Hom}(N,
- L))\) and isomorphism classes of short exact sequences
+ correspondence between elements of \(H^1(\mathfrak{g}, \operatorname{Hom}(L,
+ N))\) and isomorphism classes of short exact sequences
\begin{center}
\begin{tikzcd}
0 \rar & N \rar & M \rar & L \rar & 0
@@ -647,7 +647,7 @@ for further details.
We will use the previous result implicitly in our proof, but we will not prove
it in its full force. Namely, we will show that if \(\mathfrak{g}\) is
semisimple then \(H^1(\mathfrak{g}, M) = 0\) for all finite-dimensional \(M\),
-and that the fact that \(H^1(\mathfrak{g}, \operatorname{Hom}(N, L)) = 0\) for
+and that the fact that \(H^1(\mathfrak{g}, \operatorname{Hom}(L, N)) = 0\) for
all finite-dimensional \(N\) and \(L\) implies complete reducibility. To that
end, we introduce a distinguished element of \(\mathcal{U}(\mathfrak{g})\),
known as \emph{the Casimir element of a \(\mathfrak{g}\)-module}.