Primitive element pairs with a prescribed trace in the quartic extension of a finite field

In this paper, we give a largely self-contained proof that the quartic extension F-q4 of the finite field F-q contains a primitive element alpha such that the element alpha + alpha(-1) is also a primitive element of F-q4, and TrF (q4|Fq)(alpha) = a for any prescribed a is an element of F-q. The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [Primitive element pairs with one prescribed trace over a finite field, Finite Fields Appl. 54 (2018) 1-14.].