 Absolutely irreducible real representation

A real group representation is absolutely irreducible if its complexification is irreducible. For a finite group this is equivalent to having only scalar real endomorphisms commuting with the representation. Independent coordinate sign changes force a commuting matrix to be diagonal; a transitive cyclic permutation of the coordinates then forces its entries equal. Alternatively, the coordinate projections and the cyclic permutation directly prove irreducibility over either field.