Degree of irreducible representation over reals divides twice the group order

Statement

Suppose $G$ is a finite group and $d$ is the degree of an irreducible linear representation of $G$ over the field of real numbers $\R$. Then, $d$ divides $2|G|$, where $|G|$ denotes the order of $G$.

Facts used

1. Degree of irreducible representation divides group order (where the irreducible representation is over an algebraically closed field of characteristic zero)