  In this paper we use the $Z-$decomposition as a tool to find locally
symmetric left invariant Riemannian metrics on some Lie groups. For this
purpose, we need to compute the spectrum of the curvature operator. Since the
study of this spectrum is very difficult, we impose restriction on the class of
Riemannian Lie groups and their dimension. We investigate the $4-$dimensional
connected Riemannian Lie groups whose associated Lie algebras are
$\mathbb{A}_2\oplus 2\mathbb{A}_1,\,\mathbb{A}_{3,1}\oplus \mathbb{A}_1,\,
\mathbb{A}_{3,3}\oplus \mathbb{A}_1$ and the subclasses of $3$ and
$4-$dimensional Riemannian Lie groups which are $\mathcal{C}-$spaces.
