  It is known that every compact simple Lie group admits a bi-invariant
homogeneous Einstein metric. In this paper we use two ansatz to probe the
existence of additional inequivalent Einstein metrics on the Lie group SU (n)
for arbitrary n. We provide an explicit construction of (2k+1) inequivalent
Einstein metrics on SU (2k) and 2k inequivalent Einstein metrics on SU (2k +
1).
