  It is well known that every compact simple group manifold G admits a
bi-invariant Einstein metric, invariant under G_L\times G_R. Less well known is
that every compact simple group manifold except SO(3) and SU(2) admits at least
one more homogeneous Einstein metric, invariant still under G_L but with some,
or all, of the right-acting symmetry broken. (SO(3) and SU(2) are exceptional
in admitting only the one, bi-invariant, Einstein metric.) In this paper, we
look for Einstein metrics on three relatively low dimensional examples, namely
G=SU(3), SO(5) and G_2. For G=SU(3), we find just the two already known
inequivalent Einstein metrics. For G=SO(5), we find four inequivalent Einstein
metrics, thus extending previous results where only two were known. For G=G_2
we find six inequivalent Einstein metrics, which extends the list beyond the
previously-known two examples. We also study some cosets G/H for the above
groups G. In particular, for SO(5)/U(1) we find, depending on the embedding of
the U(1), generically two, with exceptionally one or three, Einstein metrics.
We also find a pseudo-Riemannian Einstein metric of signature (2,6) on SU(3),
an Einstein metric of signature (5,6) on G_2/SU(2)_{diag}, and an Einstein
metric of signature (4,6) on G_2/U(2). Interestingly, there are no Lorentzian
Einstein metrics among our examples.
