  We provide a new, self-contained proof of the classification of homogeneous
3-Sasakian manifolds, which was originally obtained by Boyer, Galicki and Mann.
In doing so, we construct an explicit one-to-one correspondence between simply
connected homogeneous 3-Sasakian manifolds and simple complex Lie algebras via
the theory of root systems. We also discuss why the real projective spaces are
the only non-simply connected homogeneous 3-Sasakian manifolds and derive the
famous classification of homogeneous positive quaternionic K\"ahler manifolds
due to Wolf and Alekseevskii from our results.
