  Back in 1985, Wang and Ziller obtained a complete classification of all
homogeneous spaces of compact simple Lie groups on which the standard or
Killing metric is Einstein. The list consists, beyond isotropy irreducible
spaces, of 12 infinite families (two of them are actually conceptual
constructions) and 22 isolated examples. We study in this paper the nature of
each of these Einstein metrics as a critical point of the scalar curvature
functional.
