One can prove that $\mathrm{SU}_2(\mathbb{C})/\lbrace \pm 1 \rbrace \cong \mathrm{SO}_3(\mathbb{R})$. But why we call this isomorphism exeptionnal? I believe that we call it exptionnal because there exist no $(n,k)\in\mathbb{N}^2$, and an homeomorphism $\varphi$ from $\mathrm{SU}_n$ to $\mathrm{SO}_k$.
I found a way to explain why when $k=n+1$ (which is the case in $\mathrm{SU}_2(\mathbb{C})/\lbrace \pm 1 \rbrace \cong \mathrm{SO}_3(\mathbb{R})$) using the Lie group dimension. We show that the dimension of $\mathrm{SU}_n$ is $n^2-1$ so $n^2-1=n+1$ is only possible when $n=2$.
But, in general I don't know how to proceed. Why is it not possible for exemple between $\mathrm{SU}_3$ and $\mathrm{SO}_8$ and so on?
Any help or thought will be greatly appreciate.