  We consider the moduli space of polystable $L$-twisted $G$-Higgs bundles over
a compact Riemann surface $X$, where $G$ is a real reductive Lie group, and $L$
is a holomorphic line bundle over $X$. Evaluating the Higgs field at a basis of
the ring of polynomial invariants of the isotropy representation, one defines
the Hitchin map. This is a map to an affine space, whose dimension is
determined by $L$ and the degrees of the polynomials in the basis. Building up
on the work of Kostant-Rallis and Hitchin, in this paper, as a first step in
the study of the Hitchin map, we construct a section of this map. This
generalizes the section constructed by Hitchin when $L$ is the canonical line
bundle of $X$ and $G$ is complex. In this case the image of the section is
related to the Hitchin-Teichm\"uller components of the moduli space of
representations of the fundamental group of $X$ in $G_{\mathrm{Split}}$, a
split real form of $G$. In fact, our construction is very natural in that we
can start with the moduli space for $G_{\mathrm{Split}}$, instead of $G$, and
construct the section for the Hitchin map for $G_{\mathrm{Split}}$ directly.
The construction involves the notion of maximal split subgroup of a real
reductive Lie group.
