  Let $\mathfrak g$ be a simple Lie algebra and $\mathfrak{Ab}$ the poset of
all abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. If $\mathfrak
a\in\mathfrak{Ab}$, then the normaliser of $\mathfrak a$ is a standard
parabolic subalgebra of $\mathfrak g$. We give an explicit description of the
normaliser for a class of abelian ideals that includes all maximal abelian
ideals. We also elaborate on a relationship between abelian ideals and $\mathbb
Z$-gradings of $\mathfrak g$ associated with their normalisers.
