Groups with given properties of finite subgroups

Suppose that in every finite even order subgroup $F$ of a periodic group $G$, the equality $[u,x]^2=1$ holds for any involution $u$ of $F$ and for an arbitrary element $x$ of $F$. Then the subgroup $I$ generated by all involutions in $G$ is locally finite and is a $2$-group. In addition, the normal closure of every subgroup of order $2$ in $G$ is commutative.