Suppose that $*$ is an associative and commutative binary operation on a set $S$. Let $H=\{a\in S\mid a*a=a\}$. Show that $H$ is closed under $*$.
I started this problem by listing the definitions of * being commutative and associate. So I let $a,b\in H$. I now need to show that $a*b$ is in $H$. So I said $a*b=(a*a)*(b*b)$ because of what it means to be a member of $H$. I'm not sure where to go from there or how to prove it is associative. Any help?