Introduction to Mathematical Logic, Sixth Edition
《Introduction to Mathematical Logic, Sixth Edition》 · 印刷第 17 页 Section 1.2.5「Lie Algebras Associated with Classical Groups」，典型群的定义方程微分得到对应 Lie 代数的完整段落；本书采用包括一般线性 Lie 代数和实形式的较宽口径
Following the standard convention, we have labeled each classical Lie algebra with a fraktur-font version of the name of a corresponding classical group. This passage from a Lie group to a Lie algebra, which is fundamental to Lie theory, arises by differentiating the defining equations for the group. In brief, each classical group $G$ is a subgroup of $GL(V)$ (where $V$ is a real vector space) that is defined by a set $\mathcal R$ of algebraic equations. The corresponding Lie subalgebra $\mathfrak g$ of $\mathfrak{gl}(V)$ is determined by taking differentiable curves $\sigma:(-\varepsilon,\varepsilon)\to GL(V)$ such that $\sigma(0)=I$ and $\sigma(t)$ satisfies the equations in $\mathcal R$. Then $\sigma'(0)\in\mathfrak g$, and all elements of $\mathfrak g$ are obtained in this way. This is the reason why $\mathfrak g$ is called the infinitesimal form of $G$.