[gjm11]

Does it? It looks to me as if it's saying that Clifford algebras give a better way of writing Maxwell's equations (all four of them) as a single equation with no cross products (as such) at all.

You combine the current j and charge density rho into a single four-vector current J; you combine the electric field E and magnetic field B into a single bivector F. (In spacetime, not just space, so the space of bivectors has dimension (4 choose 2) = 6 = 3+3, which is the right amount for representing E and B.) Then the Clifford-algebra analogue of the "del" operator from conventional vector calculus does div-like things and curl-like things in the right combination to encode all of the Maxwell equations.

Of course, you can do more or less the same thing without calling it Clifford algebra (or geometric algebra) and you can find it done in physics textbooks. Usually this means a whole lot of tensor formulae; see e.g. https://en.wikipedia.org/wiki/Covariant_formulation_of_class.... Geometric algebra gives a nice framework for writing it all down, though.

[jnwrd]

I apologize, my comment was too vague. My complaint is that when the connection between the Maxwell equations and Clifford algebras is first raised, there is an accompanying picture that shows the cross product in explicit coordinates.

[gjm11]

Oh, yes, I see, and I agree: that's a completely inappropriate and misleading picture.