  A well-known Peterson's theorem says that the number of abelian ideals in a
Borel subalgebra of a rank-$r$ finite dimensional simple Lie algebra is exactly
$2^r$. In this paper, we determine the dimensional distribution of abelian
ideals in a Borel subalgebra of finite dimensional simple Lie algebras, which
is a refinement of the Peterson's theorem capturing more Lie algebra
invariants.
