  We study topologically trivial $G$-Higgs bundles over an elliptic curve $X$
when the structure group $G$ is a connected real form of a complex semisimple
Lie group $G^{\mathbb{C}}$. We achieve a description of their (reduced) moduli
space, the associated Hitchin fibration and the finite morphism to the moduli
space of $G^{\mathbb{C}}$-Higgs bundles.
