  Let $G$ be an arbitrary compact Lie group. In this work we apply the method
of the analytic continuation of traces in order to compute the Wodzicki residue
for a classical pseudo-differential operator on $G$ in terms of its
matrix-valued symbol (which is globally defined on the non-commutative phase
space $G\times \widehat{G},$ with $\widehat{G}$ being the unitary dual of $G$).
Our main theorem is complementary to the results in [2], where we remove the
ellipticity hypothesis when the operators belong to the H\"ormander classes on
$G$ defined by local coordinate systems.
