Let $M^n$ be a Riemannian manifold, $G^{n + k}, \: k\geq 1 $ be a connected Lie group equipped with a left invariant metric and $f:M^n \to G^{n + k}$ one isometric immersion. Let $X, Y$ left invariant vector fields in $G$ such that in which $ p \in M $ are tangent to $ M $. Does anyone know any non trivial example where $\overline{\nabla}_{X} Y$ is also tangent to $M$ at $p$?

Here $\overline{\nabla} $ is the Riemannian connection on $G$, that for left invariant vector fields $X, Y$ is given by $$\overline{\nabla}_X Y = \frac{1}{2}\{[X, Y] - ad_{X}^{*}Y - ad_{Y}^{*}X \}$$ Thanks