2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/179808In the conventional formulation of N=1 supersymmetry, a vector multiplet is supposed to be in the adjoint representation of a given gauge group. We present a new formulation with a vector multiplet in the non-adjoint representation of SO(N) gauge group. Our basic algebra is [ T^I, T^J ] = f^{I J K} T^K, [ T^I, U^i ] = - (T^I)^{i j} U^j, [ U^i, U^j ] = - (T^I)^{i j} T^I, where T^I are the generators of SO(N), while U^i are the new 'generators' in certain non-adjoint real representation R of SO(N). We use here the word `generator' in the broader sense of the word. Such a representation can be any real representation of SO(N) with the positive definite metric, satisfying (T^I)^{i j} = - (T^I)^{j i} and (T^I)^{[ i j |} (T^I)^{| k ] l} \equiv 0. The first non-trivial examples are the spinorial 8_S and conjugate spinorial 8_C representations of SO(8) consistent with supersymmetry. We further couple the system to chiral multiplets, and show that a Higgs mechanism can give positive definite (mass)^2 to the new gauge fields for U^i. We show an analogous system working with N=1 supersymmetry in 10D, and thereby N=4 system in 4D interacting with extra multiplets in the representation R. We also perform superspace reformulation as an independent confirmation.15 pages, no figures, to appear in Phys. Rev. DHigh Energy Physics - Theorytext