Embeddings of $n$-Engel marginal subgroups in compact groups

The problem of the embedding of the $n$-Engel marginal subgroup $E^*_n(G)$ and of the $n$-Engel verbal subgroup $E_n(G)$ of a group $G$ in the terms of the lower central series and of the upper central series of $G$ seems to be still open. In full generality, some results are known when $n < 5$. Under the assumption that $G$ is a compact (Hausdorff) group, the condition $n < 5$ may be removed. This allows us to generalize classical results of Schur and Stroud.