Alternating and special orthogonal groups which are simple

I have an idle curiosity about a funny coincidence between two sequences of groups. It is well-known by those who know it well that the alternating group $A_d$ of degree $d$ is simple if and only if $d\not\in\{1,2,4\}$. By what seems like an astonishing coincidence to me, the special orthogonal group $SO(d,\mathbb{R})$ is simple if and only if $d\not\in\{1,2,4\}$. Is this merely a coincidence?

Solutions Collecting From Web of "Alternating and special orthogonal groups which are simple"