Higher order symmetric spaces and the roots of the identity in a Lie group

Let $r_k(G)$ denote the set of all $k$-roots of the identity in a Lie group $G$. We show that $r_k(G)$ is always an embedded submanifold of $G$, having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of $k$-symmetric spaces and we show, more generally, that every $k$-symmetric space of a Lie group $G$ is a covering manifold of an embedded submanifold $Orb$ of $G$. We compute also the Hessian of the inclusions of $r_k(G)$ and $Orb$ into $G$, relative to the natural connection on the domain and to the symmetric connection on $G$.