Elliptic point (awaiting review)

A transformation $(\alpha_1, \ldots, \alpha_n) \in \GL_2^+(\R)^n$ is called elliptic if each $$ \operatorname{tr}(\alpha_i)^2 - 4\det(\alpha_i) < 0. $$ If $F$ is a real quadratic field, an element $\alpha \in \GL_2^+(F)$ is called elliptic if the above condition holds for each real embedding of $F$. An elliptic point is the (necessarily unique) fixed point in $\frak{h}^2$ of an elliptic element.