We apply the algebra of compact $U(6)\ifmmode\times\else\texttimes\fi{}U(6)$, obtained from integrated current components, to meson systems. If we limit the sums over intermediate states to the lowest lying 36 $\mathrm{SU}(3)$-invariant pseudoscalar and vector meson physical states at rest, then we obtain several of the predictions of $\mathrm{SU}(6)$ models. On considering the commutation rules of magnetic-moment operators we obtain additional relations. Among these are $〈{{r}_{\ensuremath{\pi}}}^{2}〉=〈{{r}_{\ensuremath{\rho}}}^{2}〉$ and ${{\ensuremath{\mu}}_{\ensuremath{\rho}}}^{2}=\frac{1}{6}〈{{r}_{\ensuremath{\rho}}}^{2}〉$ relating the electric charge radii to the $\ensuremath{\rho}$-meson magnetic moment, with a pole model giving 2 magnetons for the $\ensuremath{\rho}$-meson total magnetic moment. We also find $\frac{〈{{r}_{\ensuremath{\pi}}}^{2}〉}{6}\ensuremath{\simeq}\frac{1.6}{{{M}_{V}}^{2}}$, using an additional consistency relation, in rough agreement with the vector-meson pole model. However, the radius of electric charge form factors is predicted to be the same as that of the divergence of the axial-vector current form factors, which disagrees with our present ideas about these form factors.