When "n" = 2"m", the situation depends more sensitively upon the parity of "m". For "m" even, a weight "λ" has an even number of minus signs if and only if −"λ" does; it follows that there are separate isomorphisms "B"±: "S"± → "S"±∗ of each half-spin representation with its dual, each determined uniquely up to scale. These may be combined into an isomorphism "B": "S" → "S"∗. For "m" odd, "λ" is a weight of "S"+ if and only if −"λ" is a weight of "S"−; thus there is an isomorphism from "S"+ to "S"−∗, again unique up to scale, and its transpose provides an isomorphism from "S"− to "S"+∗. These may again be combined into an isomorphism "B": "S" → "S"∗.