Let 𝔤 be a complex semisimple Lie algebra and θ an involutory automorphism of 𝔤. Let 𝔤 = 𝔨 ⊕ 𝔭 be the decomposition of 𝔤 where 𝔨 and 𝔭 are the respective 1 and −1 eigenspaces of θ in 𝔤. Let l be the difference of the rank of 𝔤 and that of 𝔨. The spin module S of 𝔰𝔬(𝔭) becomes a 𝔨-module via the isotropy representation ν: 𝔨 → 𝔰𝔬(𝔭). Assume that the 𝔨-module S is primary of type V ρ n . Then we prove that the multiplicity of the isotropy representation ν in V ρ n ⊗ V ρ n equals l. Let C(𝔭) be the Clifford algebra over 𝔭 and i: 𝔰𝔬(𝔭) → C(𝔭) be the natural inclusion. Let φ: 𝔨 → C(𝔭) be the composition of ν and i. We will describe the structure of the subalgebra E of C(𝔭) generated by the image of 𝔨 and also the structure of the centralizer of E in C(𝔭).