$\DeclareMathOperator\Cl{Cl}$**Problem:** Let $\Cl(d)$ be the Clifford algebra corresponding to the vector space $\mathbb{R}^d$ with the usual inner product. Given $v_1, \dotsc, v_k \in \mathbb{R}^d$, compute
the scalar part of the product $v_1 \dotsm v_k \in \Cl(d)$ in an efficient way.

One (inefficient) approach would be to first write the $v_j$ in some (ordered) orthonormal basis $\{e_i\}_{1 \leq i \leq d}$, then expand and simplify the product using that $e_i e_j = -e_j e_i$ when $i > j$ and that $e_i^2 = -1$, and finally take the scalar part (constant term) of the result. The problem with this approach is that, when expanding the product, one gets a number of monomials that is exponential in $d$, since $\Cl(d)$ is of real dimension $2^d$ with basis given by ordered products of distinct $e_i$. (Note that in my case $k$ is allowed to be of the order of $d$.)

Since I am not interested in the full product of the $v_j$ but just on the scalar part of it, I wonder if there is an efficient algorithm for its computation. So far, I have tried using some of the well known identities that hold in $\Cl(d)$, and taking a look at some of the software for doing calculations in Clifford algebras but I couldn't find this exact functionality.