Introduction to Lie Algebras and Representation Theory
《Introduction to Lie Algebras and Representation Theory》 · 印刷第 1 页基域约定及第 2—3 页 Section 1.2「Linear Lie algebras」，$C_\ell$ 的定义；保留标准交替形式的矩阵，而不以一般反对称性替代交替条件
In this chapter $F$ denotes an arbitrary (commutative) field. […] If $V$ is a finite dimensional vector space over $F$, denote by $\operatorname{End}V$ the set of linear transformations $V\to V$. […] Define a new operation $[x,y]=xy-yx$, called the bracket of $x$ and $y$. […] $C_\ell$: Let $\dim V=2\ell$, with basis $(v_1,\ldots,v_{2\ell})$. Define a nondegenerate skew-symmetric form $f$ on $V$ by the matrix $s=\begin{pmatrix}0&I_\ell\\-I_\ell&0\end{pmatrix}$. […] Denote by $\mathfrak{sp}(V)$, or $\mathfrak{sp}(2\ell,F)$, the symplectic algebra, which by definition consists of all endomorphisms $x$ of $V$ satisfying $f(x(v),w)=-f(v,x(w))$.