In the Wikipedia for the Spin group, the Clifford algebra is defined as the quotient of the tensor algebra by the ideal $v\otimes v + ||v||^2$. The reverse of an element $v$ is denoted $v^r$ and for a product of orthonormal vectors: $$(e_i\otimes e_j \otimes...\otimes e_k)^r=e_k\otimes...\otimes e_j \otimes e_i$$ They say that this operation can be extended to any element of the algebra and be made an anti-automorphism by declaring $$(a\otimes b)^r=b^r \otimes a^r$$ But doesn't this assume that $a$ and $b$ are a product of orthonormal vectors? And isn't this not true of every element of a Clifford algebra? For example, $Cl(\mathbb{R}^4)$ has $e_1e_2+e_3e_4$ as an element and, unless I'm not seeing something, can't be factored into a product of vectors.
How does the reverse act as an anti-automorphism for the algebra? Do I have a misconception about Clifford algebras or is the article inaccurate?