  This is a collection of notes on the properties of left-invariant metrics on
the eight-dimensional compact Lie group SU(3). Among other topics we
investigate the existence of invariant pseudo-Riemannian Einstein metrics on
this manifold. We recover the known examples (Killing metric and Jensen metric)
in the Riemannian case (signature (8,0)), as well as a Gibbons et al example of
signature (6,2), and we describe a new example, which is Lorentzian (ie of
signature (7,1). In the latter case the associated metric is left-invariant,
with isometry group SU(3) x U(1), and has positive Einstein constant. It seems
to be the first example of a Lorentzian homogeneous Einstein metric on this
compact manifold. These notes are arranged into a paper that deals with various
other subjects unrelated with the quest for Einstein metrics but that may be of
independent interest: Among other topics we describe the various groups that
may arise as isometry groups of left-invariant metrics on SU(3), provide
parametrizations for these metrics, give several explicit results about the
curvatures of the corresponding Levi-Civita connections, discuss modified
Casimir operators (quadratic, but also cubic) and Laplace-Beltrami operators.
In particular we discuss the spectrum of the Laplacian for metrics that are
invariant under SU(3) x U(2), a subject that may be of interest in particle
physics.
