Prove that every irreducible real representation of an abelian group is one or two dimensional.

Prove that every irreducible real representation of an abelian group is one or two dimensional.

I know the following theorem (a consequence of Schur's lemma): "Every irreducible complex linear representation of an abelian group is one-dimensional". How can I use this to prove the above?

But now I claim that every real representation arises this way: We know that every representation of $$G$$ over $$\mathbb{R}$$ or $$\mathbb{C}$$ appears inside the corresponding regular representation $$\mathbb{R}[G]$$ or $$\mathbb{C}[G]$$, but as a real representation $$\mathbb{C}[G] = \mathbb{R}[G] + i\mathbb{R}[G]$$ so every irreducible real representation appears inside the restriction of an irreducible complex representation.