Map where x*** = x* but not an involution

[icantadd]

I have been exploring an algebraic structure with a map (_)* such that
(x)*** = (x)*
but in general it is not an involution. Also, the set of elements e such that
e** = e
do not form a substructure because they are not closed to addition.

Has anyone seen such maps before, or know/can suggest a name?

Thank you!

 

[Eynstone]

How about calling this a 'hyperinvolution'?