lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -1,6 +1,6 @@
 \chapter{Semisimplicity \& Complete Reducibility}
 
-% TODO: Update the 40 pages thing when we're done 
+% TODO: Update the 40 pages thing when we're done
 Having hopefully established in the previous chapter that Lie algebras and
 their representations are indeed useful, we are now faced with the Herculean
 task of trying to understand them. We have seen that representations are a
@@ -897,10 +897,9 @@ proposition~\ref{thm:quotients-by-rads}. In practice this translates to\dots
   one-dimensional representation of \(\mathfrak{g}\).
 \end{theorem}
 
-Having achieved our goal of proving complete reducibility, we can now afford
-the luxury of concerning ourselves exclusively with irreducible
-representations. Still, our efforts towards a classification of the
-finite-dimensional representations of semisimple Lie algebras are far from
-over. In particular, there is so far no indication on how we could go about
-understanding the irreducible \(\mathfrak{g}\)-modules. Once more, we begin by
-investigating a simple case: that of \(\mathfrak{sl}_2(K)\).
+Having finally reduced our initial classification problem to that of
+classifying the finite-dimensional irreducible representations of
+\(\mathfrak{g}\), we can now focus exclusively in irreducible
+\(\mathfrak{g}\)-modules. However, there is so far no indication on how we
+could go about understanding them. In the next chapter we'll explore concrete
+examples in the hopes of finding a solution to our general problem.