An expression for $e_1e_2e_3e_4$

Question

In a real Clifford algebra $\mathbb{Cl}(2,2)$ over ${\mathbb R}^4$ with the quadratic form defined on the orthogonal basis $e_1,e_2,e_3,e_4$ by $e_1^2=1, e_2^2=1, e_3^2=-1$, and $e_4^2=-1$, find an even number of vectors $a_i$ ($i=1,2,\dots,2k$) such that $a_i^2=1$ for all $i$ and $e_1e_2e_3e_4=a_1a_2\cdots a_{2k}$.