  In the old spirit of Kaluza-Klein, we consider a spacetime of the form $P =
M_4 \times K$, where $K$ is the Lie group $\mathrm{SU}(3)$ equipped with a
left-invariant metric that is not fully right-invariant. This metric has a
${\rm U}(1) \times \mathrm{SU}(3)$ isometry group, corresponding to the
massless gauge bosons, and depends on a parameter $\phi$ with values in a
subspace of $\mathfrak{su}(3)$ isomorphic to $\mathbb{C}^2$. It is shown that
the classical Einstein-Hilbert Lagrangian density $R_P - 2 \Lambda$ on the
higher-dimensional manifold $P$, after integration over $K$, encodes not only
the Yang-Mills terms of the Standard Model over $M_4$, as in the usual
Kaluza-Klein calculation, but also a kinetic term $|{\mathrm d}^A \phi|^2$
identical to the covariant derivative of the Higgs field. For $\Lambda$ in an
appropriate range, it also encodes a potential $V(| \phi|^2)$ having absolute
minima with $|\phi_0|^2 \neq 0$, thereby inducing mass terms for the remaining
gauge bosons. The classical masses of the resulting Higgs-like and gauge bosons
are explicitly calculated as functions of the vacuum value $|\phi_0|^2$ in the
simplest version of the model. In more general versions, the classical values
of the strong and electroweak gauge coupling constants are given as functions
of the parameters of the left-invariant metric on $K$.
