diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -247,19 +247,43 @@ the Lie functor \(\mathbf{CLieGrp}_{\operatorname{simpl}} \to
\mathbb{C}\text{-}\mathbf{LieAlg}\) is also an equivalence of categories
between the category of simply connected complex Lie groups and the full
subcategory of finite-dimensional complex Lie algebras. The situation is more
-delicate in the algebraic case. For instance, given simply connected algebraic
-\(K\)-groups \(G\) and \(H\) with Lie algebras \(\mathfrak{g}\) and
-\(\mathfrak{h}\), respectively, there may be a homomorphism of Lie algebras
-\(\mathfrak{g} \to \mathfrak{h}\) which \emph{does not} come from a rational
-homomorphism \(G \to H\) -- see \cite[ch.~II]{demazure-gabriel} for instance.
-
-In other words, the Lie functor \(K\text{-}\mathbf{Grp}_{\operatorname{simpl}}
-\to K\text{-}\mathbf{LieAlg}\) fails to be full. Furthermore, there are
-finite-dimension Lie algebras over \(K\) which are \emph{not} the Lie algebra
-of an algebraic \(K\)-group, even if we allow for non-affine groups.
-Nevertheless, Lie algebras are still powerful invariants of algebraic groups.
-An interesting discussion of these delicacies can be found in sixth section of
-\cite[ch.~II]{demazure-gabriel}.
+delicate in the algebraic case. For instance, consider the complex Lie algebra
+homomorphism
+\begin{align*}
+ f : \mathbb{C} & \to \mathfrak{sl}_2(\mathbb{C}) \\
+ \lambda & \mapsto \lambda h =
+ \begin{pmatrix} \lambda & 0 \\ 0 & - \lambda \end{pmatrix}
+\end{align*}
+
+Since \(\mathfrak{sl}_2(\mathbb{C}) =
+\operatorname{Lie}(\operatorname{SL}_2(\mathbb{C}))\) and
+\(\operatorname{SL}_2(\mathbb{C})\) is simply connected, we know there exists a
+unique holomorphic group homomorphism \(g : \mathbb{C} \to
+\operatorname{SL}_2(\mathbb{C})\) between the affine line \(\mathbb{C}\) and
+the complex \emph{algebraic} group \(\operatorname{SL}_2(\mathbb{C})\) such
+that \(f = d g_1\). Indeed, this homomorphism is
+\begin{align*}
+ g : \mathbb{C} & \to \operatorname{SL}_2(\mathbb{C}) \\
+ \lambda & \mapsto \operatorname{exp}(\lambda h) =
+ \begin{pmatrix} e^\lambda & 0 \\ 0 & e^{-\lambda} \end{pmatrix},
+\end{align*}
+which is not a rational map. It then follows from the uniqueness of \(g\) that
+there is no rational group homomorphism \(\mathbb{C} \to
+\operatorname{SL}_2(\mathbb{C})\) whose derivative at the identity is \(f\).
+
+In particular, the Lie functor
+\(\mathbb{C}\text{-}\mathbf{Grp}_{\operatorname{simpl}} \to
+\mathbb{C}\text{-}\mathbf{LieAlg}\) -- between the category
+\(\mathbb{C}\text{-}\mathbf{Grp}_{\operatorname{simpl}}\) of complex algebraic
+groups and the category of complex Lie algebras -- fails to be full. Similarly,
+the functor \(\mathbb{C}\text{-}\mathbf{Grp}_{\operatorname{simpl}} \to
+\mathbb{C}\text{-}\mathbf{LieAlg}\) is \emph{not} essentially surjective onto
+the subcategory of finite-dimensional algebras: every finite-dimensional
+complex Lie algebra is isomorphic to the Lie algebra of a unique simply
+connected complex Lie group, but there are simply connected complex Lie groups
+which are not algebraic groups. Nevertheless, Lie algebras are still powerful
+invariants of algebraic groups. An interesting discussion of some of these
+delicacies can be found in sixth section of \cite[ch.~II]{demazure-gabriel}.
All in all, there is a profound connection between groups and
finite-dimensional Lie algebras throughout multiple fields. While perhaps
@@ -499,7 +523,7 @@ semisimple and reductive algebras by modding out by certain ideals, known as
\emph{radicals}.
\begin{definition}\index{Lie algebra!radical}
- Let \(\mathfrak{g}\) be a finite-dimensioanl Lie algebra. The sum
+ Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra. The sum
\(\mathfrak{a} + \mathfrak{b}\) of solvable ideals \(\mathfrak{a},
\mathfrak{b} \normal \mathfrak{g}\) is again a solvable ideal. Hence the sum
of all solvable ideals of \(\mathfrak{g}\) is a maximal solvable ideal, known
@@ -780,7 +804,7 @@ definition.
\end{definition}
Hence there is a one-to-one correspondence between representations of
-\(\mathfrak{g}\) and \(\mathcal{U}(\mathfrak{g})\)-modules.
+\(\mathfrak{g}\) and \(\mathcal{U}(\mathfrak{g})\)-modules.
\begin{example}\index{\(\mathfrak{g}\)-module!trivial module}
Given a Lie algebra \(\mathfrak{g}\), the zero map \(0 : \mathfrak{g} \to K\)