This paper investigates the d = 4, 𝒩=4 Abelian, global Super-Yang Mills system (SUSY-YM). It is shown how the 𝒩=2 Fayet Hypermultiplet (FH) and 𝒩=2 vector multiplet (VM) are embedded within. The central charges and internal symmetries provide a plethora of information as to further symmetries of the Lagrangian. Several of these symmetries are calculated to second order. It is hoped that investigations such as these may yield avenues to help solve the auxiliary field closure problem for d = 4, 𝒩=4, SUSY-YM and the d = 4, 𝒩=2 Fayet-Hypermultiplet, without using an infinite number of auxiliary fields.