Are there nice Lie groups $G$ and $H$ such that $$G \times G \times G /\mathbb Z_3 \cong H?$$
I'm looking for analogies to $SO(3)\times SO(3) /\mathbb Z_2 \cong SO(4)$, which has an interpretation in that left- and right-isoclinic rotations provide a double cover of $SO(4)$.
I can see solutions that aren't particularly 'nice'; eg for any $G$ with a $3$-element normal subgroups $\mathbb Z_3 \sim N \unlhd G$ we could define $H = (G/\mathbb Z_3) \times G \times G$. (An example would be $G = U(1), N = \left\{1, e^{\tfrac 23\pi i}, e^{\tfrac 43\pi i}\right\}$.)
I'm hoping there are constructions that preserve a symmetry between the $3$ copies of $G$.