Lie algebra of a Lie group

Let $G$ be a Lie group with identity element $e$. Its Lie algebra is the real vector space

To define the Lie bracket on $\mathfrak{g}$, use left translations . For each $X\in T_eG$, there is a unique left-invariant vector field $\widetilde X$ on $G$ satisfying $\widetilde X_e=X$, namely

The bracket on $\mathfrak{g}$ is then defined by

where $[\widetilde X,\widetilde Y]$ is the Lie bracket of vector fields on $G$. This operation is bilinear, alternating, and satisfies the Jacobi identity, making $\mathfrak{g}$ into a Lie algebra in the usual algebraic sense.

Examples

1. General linear group. For $G=\mathrm{GL}(n,\mathbb{R})$, one has $\mathfrak{g}=\mathfrak{gl}(n,\mathbb{R})$ (all $n\times n$ real matrices) with bracket $[A,B]=AB-BA$.
2. Special orthogonal group. For $G=\mathrm{SO}(n)$, the Lie algebra is $\mathfrak{so}(n)=\{A\in\mathfrak{gl}(n,\mathbb{R})\mid A^\top=-A\}$ with the same commutator bracket.
3. Abelian Lie groups. For $G=\mathbb{R}^n$ under addition (or a torus $T^n$), the bracket on $\mathfrak{g}\cong\mathbb{R}^n$ is identically zero.