Lie algebra of a matrix Lie group

For this sub-case, we can define the Lie algebra of a Lie group $G$ as the set of all matrices $M\in G$ such that for all $t\in \mathbb{R}$: If we fix a given $M$ and vary $t$, we obtain a subgroup of $G$. This type of subgroup is known as a one parameter subgroup.

The immediate question is then if every element of $G$ can be reached in a unique way (i.e. is the exponential map a bijection). By looking at the matrix logarithm however we conclude that this is not the case for real matrices, but it is for complex matrices.

Examples:

TODO example it can be seen that the Lie algebra is not closed matrix multiplication, even though the corresponding group is by definition. But it is closed under the Lie bracket operation.

This makes it clear how the Lie bracket can be seen as a "measure of non-commutativity"

Because the Lie bracket has to be a bilinear map, all we need to do to specify it uniquely is to specify how it acts on every pair of some basis of the Lie algebra.

Then, together with the Baker-Campbell-Hausdorff formula and the Lie group-Lie algebra correspondence, this forms an exceptionally compact description of a Lie group.

The one parameter subgroup of a Lie group for a given element $M$ of its Lie algebra is a subgroup of $G$ given by:

Intuitively, $M$ is a direction, and $t$ is how far we move along a given direction. This intuition is especially vivid in for example in the case of the Lie algebra of $SO(3)$, the rotation group.

One parameter subgroups can be seen as the continuous analogue to the cycle of an element of a group.