Is every abelian group $A$ where every element has order two isomorphic to a direct product of cyclic groups of order two, $A\cong C_2\times C_2\times\ldots$?
I ask because I used this "fact" in one of my old answers here (which is relevant to some work I am doing), and have just realised that this is not obvious, and so perhaps not true.
Am I perhaps just not seeing something which I thought was obvious at the time? Or is there something more subtle going on?
(Note that there is no assumption that $A$ is finitely generated.)