  In this paper we give a spinorial representation of submanifolds of any
dimension and codimension into Lie groups equipped with left invariant metrics.
As applications, we get a spinorial proof of the Fundamental Theorem for
submanifolds into Lie groups, we recover previously known representations of
submanifolds in $\mathbb{R}^n$ and in the 3-dimensional Lie groups $S^3$ and
$E(\kappa,\tau),$ and we get a new spinorial representation for surfaces in the
3-dimensional semi-direct products: this achieves the spinorial representations
of surfaces in the 3-dimensional homogeneous spaces. We finally indicate how to
recover a Weierstrass-type representation for CMC-surfaces in 3-dimensional
metric Lie groups recently given by Meeks, Mira, Perez and Ros.
