The motivation for my question comes from the definition of rank of a given globally symmetric space: it is based on the image of a maximal abelian subalgebra of a given algebra by the exponential map. I do not wish to enter in the details of this, since I want the answer to be more general as possible.
My question is: Is there a geometric interpretation for a subalgebra to be abelian? (Take the Lie algebra of a Lie group)
The Lie algebra of a compact, connected Lie group $G$ is abelian if and only if $G$ is a torus (that is, if and only if $G\cong \mathbb{S}^1\times\cdots\times \mathbb{S}^1$ is a finite product of circle groups); to prove this, note that the exponential map is the universal cover. So, abelian Lie subalgebras of the Lie algebra of a Lie group give rise to embedded tori in the Lie group. A maximal abelian subalgebra gives rise to a maximal torus, which is quite crucial in the representation theory of compact Lie groups.
Hope this helps! Please let me know if you would like me to elaborate further.