lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -170,38 +170,32 @@ this last construction.
   The Lie algebra of the affine algebraic group
   \[
     \operatorname{Sp}_{2 n}(K)
-    = \left \{ M \in \operatorname{SL}_{2 n}(K) :
-        M
-        \begin{pmatrix}
-          0                     & \operatorname{Id}_n \\
-          - \operatorname{Id}_n & 0
-        \end{pmatrix}
-        M^{-1} =
-        \begin{pmatrix}
-          0                     & \operatorname{Id}_n \\
-          - \operatorname{Id}_n & 0
-        \end{pmatrix}
-      \right \}
+    = \{
+        g \in \operatorname{GL}_{2 n}(K) :
+        \omega(g \cdot v, g \cdot w) = \omega(v, w) \, \forall v, w \in K^{2n}
+      \}
   \]
   is canonically isomorphic to the Lie algebra
   \[
     \mathfrak{sp}_{2 n}(K) =
     \left\{
-      X \in \mathfrak{gl}_{2 n}(K) : X^\top
       \begin{pmatrix}
-        0                     & \operatorname{Id}_n \\
-        - \operatorname{Id}_n & 0
+        X &  Y      \\
+        Z & -X^\top
       \end{pmatrix}
-      = -
-      \begin{pmatrix}
-        0                     & \operatorname{Id}_n \\
-        - \operatorname{Id}_n & 0
-      \end{pmatrix}
-      X
+      : X, Y, Z \in \mathfrak{gl}_n(K), Y = Y^\top, Z = Z^\top
     \right\},
   \]
   with bracket given by the usual commutator of matrices -- where
-  \(\operatorname{Id}_n\) denotes the \(n \times n\) identity matrices.
+  \[
+    \omega(
+      (v_1, \ldots, v_n, \dot v_1, \ldots, \dot v_n),
+      (w_1, \ldots, w_n, \dot w_1, \ldots, \dot w_n)
+    )
+    = v_1 \dot w_1 + \cdots + v_n \dot w_n
+    - \dot v_1 w_1 - \cdots - \dot v_n w_n
+  \]
+  is, of course, the standard symplectic form of \(K^{2n}\).
 \end{example}
 
 It is important to point out that the construction of the Lie algebra