Let X be a compact Riemann surface of genus g ≥ 2 and M(E6) be the moduli space of E6-Higgs bundles over X. We consider the automorphisms σ+ of M(E6) defined by σ (E, φ) = (E∗ , −φt) , induced by the action of the outer involution of E6 in M(E6), and σ− defined by σ (E, φ) = (E* , -φt), which results from the combination of σ+ with the involution of M(E6), which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ+ and σ−, as F4-Higgs bundles, F4-Higgs pairs associated with the fundamental irreducible representation of F4, and PSp(8, C)-Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp(8, C). Finally, we describe the reduced notions of semistability and polystability for these objects. Citation: Antón-Sancho, Álvaro. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface. Open Mathematics, vol. 20, no. 1, 2022, pp. 1723-1733. https://doi.org/10.1515/math-2022-0543