Let $n\in\mathbb N$, let $M(n,\mathbb C)$ denote the real [vector space](/page/Vector%20Space) of complex $n\times n$ matrices, let $GL(n,\mathbb C)$ be its group of invertible elements, and let $I\in GL(n,\mathbb C)$ be the identity matrix. Let $G\le GL(n,\mathbb C)$ be a subgroup equipped with a smooth embedded submanifold structure inside $M(n,\mathbb C)$ such that multiplication and inversion on $G$ are smooth. For $X\in M(n,\mathbb C)$, define $\exp X$ by

\begin{align*} \exp X:=\sum_{k=0}^{\infty}\frac{X^k}{k!}. \end{align*}

Define the [Lie algebra](/page/Lie%20Algebra) of $G$ by one-parameter subgroups:

\begin{align*} \mathfrak g:=\{X\in M(n,\mathbb C):\exp(tX)\in G\text{ for every }t\in\mathbb R\}. \end{align*}

Then, under the identification of $T_I G$ with the real linear subspace $dj_I(T_I G)\subset M(n,\mathbb C)$ induced by the inclusion map $j:G\hookrightarrow M(n,\mathbb C)$ and the standard translation identification $T_I M(n,\mathbb C)\cong M(n,\mathbb C)$,

\begin{align*} \mathfrak g=T_I G. \end{align*}

Consequently,

\begin{align*} \dim_{\mathbb R}\mathfrak g=\dim_{\mathbb R}G. \end{align*}