On periodic completely splittable groups

We study an infinite periodic group $G$ with involutions that coincides with the set-theoretic union of a collection of proper locally cyclic subgroups with trivial pairwise intersections. It is proved that if $G$ contains an elementary subgroup $E_8$, then either $G$ is locally finite (and its structure is described) or its subgroup $O_2(G)$ is elementary and strongly isolated in $G$. If $G$ has a finite element of order greater than 2 and the $2$-rank of $G$ is not $2$, then $G$ is locally finite, and its structure is described.