J_{\pm} = J_{1} \pm i J_{2}

. The SU(2) Lie algebra relations imply

[J_{3}, J_{\pm}] = \pm J_{\pm}, [J_{+}, J_{-}] = 2 J_{3} .

Thus

d (J_{\pm}) v_{m}

m \pm 1

when nonzero. Starting from a maximum-weight vector

v_{I}

, repeated lowering gives weights

I, I - 1, \dots, - I .

∥ J_{-} ∣ I, m ⟩ ∥^{2} = (I + m) (I - m + 1),

shows that lowering stops precisely at

m = - I

. Therefore

2 I

2 I + 1

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