Let $A$ be a finite dimensional real algebra, and let $F:A\times A\to A$ be the multiplication map $F(a,b)=ab$. Is it true that the map $F$ is smooth? (Here I am considering the canonical smooth structure of $A$: choose a real linear basis $v_1,\dots,v_n$ of $A$ and use this basis to linearly identify $A$ with $\Bbb R^n$; this gives a smooth structure on $A$, and it is independent of the choice of the basis.)
P.S. Actually I want the result when $A$ is the Clifford algebra $Cl(V)$ of a finite dimensional inner product space $V$.