Abelian Lie algebra

Definition
A Lie algebra is said to be Abelian if the Lie bracket of any two elements in it is zero.

Lie algebra of an Abelian Lie group is Abelian
If we start with an Abelian Lie group, then its Lie algebra is also Abelian. This follows from the fact that for a Lie group, the Lie bracket of two tangent vectors $$v$$ and $$w$$ is the limit of a certain difference expression of conjugates.

More explicitly, there is a natural action of the Lie group on its Lie algebra, called the adjoint representation, and the commutator $$[v,w]$$ can be defined as the differential at $$t=0$$ of $$\exp{tv}.w - w$$. When the group is Abelian, the adjoint representation is trivial so $$\exp{tv}.w = w$$ and hence $$\exp{tv}.w - w = 0$$ identically. Thus $$[v,w]=0$$ (as the differential of a constant function).

If the Lie algebra is Abelian then the group is nearly so
The Lie algebra being Abelian does not force the group to be Abelian. However, we can certainly conclude that the connected component containing the identity is Abelian. In other words, the group is an Abelian-by-discrete group.

Weaker properties

 * Nilpotent Lie algebra
 * Solvable Lie algebra