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