lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -2,7 +2,11 @@
 
 \epigraph{Nobody has ever bet enough on a winning horse.}{Some gambler}
 
+% TODOOO: Point out we are now working with finite-dimensional Lie algebras
+% over an algebraicly closed field of characteristic zero
+
 % TODO: Update the 40 pages thing when we're done
+% TODO: Have we seen the fact representations are useful?
 Having hopefully established in the previous chapter that Lie algebras are
 indeed useful, we are now faced with the Herculean task of trying to
 understand them. We have seen that representations are a remarkably effective
@@ -12,18 +16,13 @@ Lie algebra? This is a question that have sparked an entire field of research,
 and we cannot hope to provide a comprehensive answer the 40 pages we have left.
 Nevertheless, we can work on particular cases.
 
-Like any sane mathematician would do, we begin by studying a simpler case. The
-restrictions we impose are twofold: restrictions on the algebras whose
-representations we'll classify, and restictions on the representations
-themselves. As indicated by the title of this chapter, we will initially focus
-on the \emph{semisimple} Lie algebras algebras. We will later relax this
-restriction a bit in chapter~\ref{ch:mathieu} when we dive into
-\emph{reductive} Lie algebras. The first question we need to answer is: why are
-semisimple algebras simpler -- or perhaps \emph{semisimpler} -- to understand
-than any old Lie algebra? Well, the special thing about semisimple algebras is
-that the relationship between their indecomposable representations and their
-irreducible representations is much clearer -- at least in finite dimension.
-Namely\dots
+Like any sane mathematician would do, we begin by studying a simpler case,
+which is that of \emph{semisimple} Lie algebras algebras. The first question we
+have is thus: why are semisimple algebras simpler -- or perhaps
+\emph{semisimpler} -- to understand than any old Lie algebra? Well, the special
+thing about semisimple algebras is that the relationship between their
+indecomposable representations and their irreducible representations is much
+clearer -- at least in finite dimension. Namely\dots
 
 \begin{proposition}\label{thm:complete-reducibility-equiv}
   Given a finite-dimensional Lie algebra \(\mathfrak{g}\) over \(K\), the