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For G = SU(n), Sp(n) or Spin(n), let $C_G\left(SU\left(2\right)\right)$ be the centralizer of a certain SU(2) in G. We have a natural map $J:G/C_G\left(SU\left(2\right)\right)\to \Omega ₀³G$. For a generator α of $H⁎(G/C_G\left(SU\left(2\right)\right);\mathbb{Z}/2)$, we describe J⁎(α). In particular, it is proved that $J⁎:H⁎(G/C_G\left(SU\left(2\right)\right);\mathbb{Z}/2)\to H⁎(\Omega ₀³G;\mathbb{Z}/2)$ is injective.