  The paper surveys recent progress in the search for an appropriate internal
space algebra for the Standard Model (SM) of particle physics. As a starting
point serve Clifford algebras involving operators of left multiplication by
octonions. A central role is played by a distinguished complex structure which
implements the splitting of the octonions ${\mathbb O} = {\mathbb C} \oplus
{\mathbb C}^3$ reflecting the lepton-quark symmetry. Such a complex structure
in $C\ell_{10}$ is generated by the $C\ell_6(\subset C\ell_8\subset
C\ell_{10})$ volume form, $\omega_6 = \gamma_1 \cdots \gamma_6$, left invariant
by the Pati-Salam subgroup of $Spin(10)$, $G_{\rm PS} = Spin (4) \times Spin
(6) / {\mathbb Z}_2$. While the $Spin(10)$ invariant volume form
$\omega_{10}=\gamma_1 ... \gamma_{10}$ is known to split the Dirac spinors of
$C\ell_{10}$ into left and right chiral (semi)spinors, ${\cal P} = \frac12 (1 -
i\omega_6)$ is interpreted as the projector on the 16-dimensional
\textit{particle subspace} (annihilating the antiparticles). The standard model
gauge group appears as the subgroup of $G_{PS}$ that preserves the sterile
neutrino (identified with the Fock vacuum). The $\mathbb{Z}_2$-graded internal
space algebra $\mathcal{A}$ is then included in the projected tensor product:
$\mathcal{A}\subset \mathcal{P}C\ell_{10}\mathcal{P}=C\ell_4\otimes \mathcal{P}
C\ell_6^0\mathcal{P}$. The Higgs field appears as the scalar term of a
superconnection, an element of the odd part, $C\ell_4^1$, of the first factor.
The fact that the projection of $C\ell_{10}$ only involves the even part
$C\ell_6^0$ of the second factor guarantees that the colour symmetry remains
unbroken. As an application we express the ratio $\frac{m_H}{m_W}$ of the Higgs
to the $W$-boson masses in terms of the cosine of the {\it theoretical}
Weinberg angle.
