The Lie subgroup of the compact Lie group

Question

$G$ is a compact connected Lie group with Lie algebra $g$ whose center is $h$. Let $h^{\bot}$ be the orthogonal complement of $h$ where the inner product is chosen to be invariant under the adjoint representation of $G$. If $K$ is the connected Lie subgroup associated with $h^{\bot}$, then is $K$ compact?