Is it possible to complete a basis for a finite dimensional free module over a finite dimensional associative unital real algebra?

Question

Let $\mathbb F$ be a finite-dimensional associative unital real algebra. Let $V:=\mathbb F^n$ and $p_1 \in V$ be a vector such that $xp_1=0$ only has $x=0$ as solution.

Question: Are there $p_2,\ldots, p_n \in V$ such that $p_1,\ldots,p_n$ is a basis for $V$?

The result is true for $\mathbb F$ commutative and for the quaternions. Nevertheless, it does seem that the result holds in general.