  A Moore--Penrose inverse of an arbitrary complex matrix A is defined as a
unique matrix A' such that AA'A=A, A'AA'=A', and AA', A'A are Hermite matrices.
We show that this definition has a natural generalization in the context of
shortly graded simple Lie algebras corresponding to parabolic subgroups with
aura (abelian unipotent radical) in simple complex Lie groups, or equivalently
in the context of simple complex Jordan pairs. We give further generalizations
and applications.
