  We study semi-Riemannian submanifolds of arbitrary codimension in a Lie group
$G$ equipped with a bi-invariant metric. In particular, we show that, if the
normal bundle of $M \subset G$ is closed under the Lie bracket, then any normal
Jacobi operator $K$ of $M$ equals the square of the associated invariant shape
operator $\alpha$. This permits to understand curvature adaptedness to $G$
geometrically, in terms of left translations. For example, in the case where
$M$ is a Riemannian hypersurface, our main result states that the normal Jacobi
operator commutes with the ordinary shape operator precisely when the
left-invariant extension of each of its eigenspaces has first-order tangency
with $M$ along all the others. As a further consequence of the equality $K =
\alpha^{2}$, we obtain a new case-independent proof of a well-known fact: every
three-dimensional Lie group equipped with a bi-invariant semi-Riemannian metric
has constant curvature.
