lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -509,7 +509,10 @@
   The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
   from corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
   corollary that \strong{(iii)} implies \strong{(ii)}. All it's left is to show
-  \strong{(ii)} implies \strong{(iii)}.
+  \strong{(ii)} implies \strong{(iii)}\footnote{This isn't already clear from
+  corollary~\ref{thm:cuspidal-mod-equivs} because, at first glance,
+  $\mathcal{M}[\lambda]$ may not be irreducible for some $\lambda$ satisfying
+  \strong{(ii)}. We will show this is never the case.}.
 
   Suppose \(F_\alpha\) acts injectively in the subrepresentation
   \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since