Let X be a compact Riemann surface of genus \(g\ge 2\) . The non-trivial outer involution of \(E_6\) acts on the moduli space \(\mathcal {M}(E_6)\) of polystable \(E_6\) -Higgs bundles over X, leaving fixed those \(E_6\) -Higgs bundles that reduce their structure group to the complex subgroup \(F_4\) . Then, the image of the forgetful map \(\mathcal {F}:\mathcal {M}(F_4)\rightarrow \mathcal {M}(E_6)\) induced by the inclusion \(F_4\hookrightarrow E_6\) is composed of fixed points of the outer involution action. In this work, it is proved, as the main result, that \(\mathcal {F}\) is an injective morphism, which implies that \(\mathcal {M}(F_4)\) is a subvariety of \(\mathcal {M}(E_6)\) . During the proof process, it is checked that the algebra of \(F_4\) -invariant polynomials is isomorphic to the subalgebra of \(E_6\) -invariant polynomials fixed by the action of the outer involution of \(E_6\) . As an application, it is proved that the restriction of \(\mathcal {F}\) to any fiber of the Hitchin fibration associated to \(\mathcal {M}(F_4)\) is a closed embedding and that the morphism \(\mathcal {F}\) is indeed a closed embedding, so \(\mathcal {M}(F_4)\) is a closed subvariety of \(\mathcal {M}(E_6)\) . Some implications of the above results are provided involving the computation of Pontryagin and Chern classes and Betti numbers of \(\mathcal {M}(F_4)\) through the morphism \(\mathcal {F}\) . Moreover, the nature of this embedding is further investigated by analyzing the intersection behavior of Hitchin fibers, studying the discriminant loci, and computing the normal bundle and its Chern class. Additionally, the embedding is shown to optimally respect the hyperkähler structure, with tangent and normal spaces being symplectically orthogonal.
