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This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise. Let $\mathfrak{g}$ denote a complex simple Lie algebra of ...

The exceptional complex simple Lie algebra $F_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in ...