Curvature adapted submanifolds of bi-invariant Lie groups

We study submanifolds of arbitrary codimension in a Lie group $\mathsf{G}$ equipped with a bi-invariant metric. In particular, we show that, if the normal bundle of $M \subset \mathsf{G}$ is abelian, then the normal Jacobi operator of $M$ equals the square of its invariant shape operator. This allows us to obtain geometric conditions which are necessary and sufficient for the submanifold $M$ to be curvature adapted to $\mathsf{G}$.