Semisimple extensions of the Standard Model gauge algebra

We show how one may classify all semisimple algebras containing the $\mathfrak{su}(3)\oplus \mathfrak{su}(2) \oplus \mathfrak{u}(1)$ symmetry of Standard Model and acting on some given matter sector, enabling theories beyond with unification (partial or total) symmetries (gauge global) to be catalogued. With just a single generation fermions plus singlet neutrino, only {gauge} correspond well-known $\mathfrak{su}(5),\mathfrak{so}(10),$ $\mathfrak{su}(4)\oplus \mathfrak{su}(2)$, but two more generations limited number exotic mixing flavour, colour, electroweak degrees freedom become possible. provide complete catalogue in case 3 fewer outline our method generalizes cases additional matter.