SU(2) and SO(3) share roots but are not isomorphic

Given: Assume the Axiom of Choice; the double cover $SU(2)\to SO(3)$ with kernel $\{\pm I\}$.

$SU(2)$ and $SO(3)$ have the same $A_1$ root system and are centrally isogenous, the adjoint double cover being $SU(2)\to SO(3)=SU(2)\mathrm{/}\{\pm I\}$ (Semisimple compact groups up to isogeny, Character lattices of SU(2) and SO(3)).

The centre of $SU(2)$ is $\{\pm I\}$: a central matrix commutes with every $\mathrm{diag}(z,z^{-1})$, so it is diagonal, and commuting with $J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ makes its diagonal entries equal; determinant one then gives $\pm I$. The centre of $SO(3)$ is trivial: commuting with the three coordinate half-turns makes a central rotation diagonal, and commuting with the cyclic coordinate permutation makes its three diagonal entries equal; orthogonality and determinant one then force $I$. Finally, if $\phi :G\to H$ is an isomorphism and $z\in Z(G)$, then $\phi (z)$ commutes with every $\phi (g)\in H$, so $\phi (Z(G))=Z(H)$. [matrix multiplication, group axioms]

The character lattice of $SU(2)$ is $P=\mathbb{Z}\omega$ while that of $SO(3)$ is $Q=2\mathbb{Z}\omega$ (Character lattices of SU(2) and SO(3)).