I found that all the commutative rings with involution I know are the following:
- complex number with complex conjugation (plus similar constructions based on rationals and its extensions),
- any commutative ring with trivial involution,
- direct sum of two copies of a ring with involution * (not necessary nontrivial) with involution which sends pair $(a,b)$ to $(b^*,a^*)$,
- direct sum of involutionary rings with involution acting component-wise.
My question is: are there any other commutative rings with involution?