Prove that every multivector which does not have an inverse has an idempotent for a factor.
Define an idempotent as a multivector $A$ with the property that $A^2=A$ and $A \neq 1$.
I can show it for specific cases, such as, $B = \beta + \mathbf b$, $\beta$ a scalar and $\mathbf b$ a vector, and $C = \langle C \rangle_0 + \langle C \rangle_1 + \langle C \rangle_2$. But I can't figure out how to show it in the general case.
A geometric algebra over a nonsingular finite dimensional $F$-vector space is a semisimple ring, so it is von Neumann regular. As such, for every $a$ in the ring, there exists $x$ such that $axa=a$.
From this you can compute that $ax$ is an idempotent, and clearly it's a factor of $a$. If $ax=1$, then since the ring is Artinian it is also true that $xa=1$, and $a$ would be a unit. So if $a$ is not a unit, $ax\neq 1$.