SU(2) and SO(3): same local Lie theory, different groups

For a unit quaternion $a$, the map $u\mapsto au{a}^{-1}$ preserves the norm and orientation on $\mathrm{Im}\mathbb{H}$, so it gives $q(a)\in \mathrm{SO}(3)$. It is a homomorphism. If it fixes every imaginary quaternion, then $a$ commutes with $i,j,k$, hence is real; unit length gives $a=\pm 1$. Thus $\ker q=\{\pm 1\}$.

Differentiating at $1$ sends an imaginary quaternion $u$ to $A_u(v)=uv-vu=2u\times v$. This map is injective and both real vector spaces have dimension three, so it is an isomorphism. Its bracket compatibility follows by differentiating the homomorphism. The image of $q$ is therefore an open subgroup of connected $\mathrm{SO}(3)$, hence all of it; $q$ is a two-sheeted covering.

Since $\mathrm{SU}(2)\cong S^3$ is simply connected, [L2] identifies $q$ as the universal cover. Its deck group is its kernel, so ${\pi }_1(\mathrm{SO}(3))\cong \{\pm 1\}\cong \mathbb{Z}\mathrm{/}2$. A Lie-group isomorphism is a diffeomorphism and would preserve the fundamental group; hence the groups are not isomorphic even though step 2.1 gives isomorphic Lie algebras. This realizes [L1].

The quaternionic calculation itself is finite. The declared ${\mathsf{A}\mathsf{C}}_{\omega }$ is propagated from the library's covering-group suppliers [L1]–[L2]; it is not silently strengthened and no additional choice is used here.