Let $k$ be a totally real number field. Every quaternion algebra $B$ over $k$ can be written as $$B = \left(\frac{a,b}{k}\right), $$ for some constants $a,b \in k^\times$. My question is, can I always choose these constants so that $\sigma(a) > 0$ for all real embeddings $\sigma \colon k \hookrightarrow \mathbb{R}$ at which $B$ splits?

A priori, one only has $\sigma(a) > 0$ or $\sigma(b) > 0$ for each of the embeddings $\sigma \colon k \hookrightarrow \mathbb{R}$ which split $B$. But in many examples one can use the symmetries $\bigl(\frac{a,\,b}{k}\bigr) = \bigl(\frac{b,\,a}{k}\bigr) = \bigl(\frac{a,\,-ab}{k}\bigr)$ to achieve the above condition.