Orbits of unipotent elements in the generalized symmetric space defined by inner involutions of SL3(𝔽q) and SL4(𝔽q) for q = 2n

In this paper, we characterize and classify the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of [Formula: see text] and [Formula: see text] where [Formula: see text] is a finite field of even characteristic. We also provide some generalized results for [Formula: see text]. These results, together with our previous work that determined similar orbits when [Formula: see text] is a finite field of odd characteristic, complete the classification of the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of [Formula: see text] and [Formula: see text] for any finite field [Formula: see text] of any characteristic.