Let $k$ be a field, and $A$ an associative $k$-algebra with an identity element. Say that $A$ is quadratic if any subalgebra of $A$ generated by a single element has dimension at most two.

I am looking for a reference for the following (or any similar) result:

Assume that ${\rm char}(k)\ne2$, and let $A$ be a quadratic division algebra over $k$ with $\dim_k(A)\ge3$. Then $A$ is a quaternion algebra.

(I expect this to be well known, but am not aware of any reference.)