A Lie algebra is abelian if its bracket is identically 0, in that for all we have
[x,y] = 0 \,.
Every vector space has a (necessarily unique) abelian Lie algebra structure. As such, we may identify an abelian Lie algebra with its underlying vector space.
A -dimensional or -dimensional Lie algebra must be abelian. The -dimensional Lie algebra is the trivial Lie algebra. The -dimensional Lie algebra is a simple object in LieAlg, but it is traditionally not considered a simple Lie algebra.