It is well known that the Standard Model (SM) gauge group is a subgroup of SU(5): SU(3)×SU(2)×U(1) ⊂ SU(5) This can be easily checked using the method of Dynkin diagrams. Is this subgroup an invariant subgroup such that, gSU(5)gSMgSU(5)=g′SM, where gSU(5) (gSM) is an element of SU(5) (SM)?
Background: The reason I'm interested in this is because then its necessarily true that the non-SM gauge group generators of SU(5) can be written as solely off-diagonal matrices and the SM as solely diagonal (this is easy to see by writing the matrices in block diagonal form), which simplifies calculations.
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