Let $G$ be a compact connected simple Lie group. It is known that its third homotopy group $\pi_3(G)$ is isomorphic to $\mathbb{Z}$. More precisely, there is a Lie group homomorphism $$\rho:SU(2)\longrightarrow G$$ which induces an isomorphism $$\rho_*:\pi_3(SU(2))\overset{\cong}{\longrightarrow}\pi_3(G).$$ (Recall that $SU(2)\cong S^3$ so $\pi_3(SU(2))=\Bbb Z$.)

Clearly, all conjugates of $\rho$ have the same property. Is there a Lie group homomorphism $\varphi:SU(2)\to G$ such that $\varphi_*=\rho_*$ but $\varphi$ is not conjugate to $\rho$?