= Distinct-character extensions of an abelian group split
A two-dimensional representation of an <abelian group> fitting into an extension of two distinct <linear characters> splits. Choose an element on which the <linear characters> differ. Its matrix has two distinct <eigenvalues>, and all commuting group operators preserve its two eigenlines. Those eigenlines supply the two invariant <linear character> summands. The argument does not apply when the two <linear characters> coincide.
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