Lie algebra of a real Lie group

Definition
Suppose $$G$$ is a real Lie group, i.e., a Lie group over the field of real numbers $$\R$$. The Lie algebra of $$G$$ is a Lie algebra over $$\R$$ defined as follows: it is the collection of all left-$$G$$-invariant derivations on $$G$$ (viewed as a differential manifold). The structures are described in the table below:

Although the definition here uses left invariance, an equivalent definition using right invariance gives the same Lie algebra because every Lie group is naturally isomorphic to its opposite Lie group via the inverse map (a Lie group version of the fact that every group is naturally isomorphic to its opposite group via the inverse map).

Related notions

 * Lie algebra of an algebraic group: This notion is defined for an algebraic group over any field. It turns out that over the field of real numbers, the Lie algebra of an algebraic group coincides with the Lie algebra of the corresponding real Lie group.
 * Lie algebra of a formal group law