Lemma 7. The group G_c is semisimple. At connected monodromy level the Hodge circles normalize G_c , are locally conjugate by G_c(ℝ)⁺ , and have locally constant action on every G_c -invariant tensor space. The horizontal period …

André's monodromy normality theorem applies to this polarizable pure variation over a smooth connected algebraic variety: the connected monodromy is normal in the derived generic Mumford--Tate group. It is therefore semisimple. The same statement applies after a finite étale cover and after restriction to a smooth algebraic subvariety, with its own generic Mumford--Tate group. The theorem of the fixed part for polarized variations says that the invariant part of each tensor variation is a constant Hodge structure; see . These tensor invariants determine the reductive group G_c . To see the latter assertion in the form needed here, a G_c -stable subspace in a tensor representation has a G_c -equivariant complementary subspace, so it can be encoded by its invariant projector. The tensor-stabilizer description of an algebraic subgroup then applies . Thus, in a fixed flat frame, the Hodge circles h_x normalize G_c , and h_x(z)h_ x_0 (z)^ -1 ∈G_c(ℝ). In particular their images in the quotient by G_c are constant. For completeness, local conjugacy follows directly from compactness of the circle. Along a smooth path of these homomorphisms, the logarithmic derivative is a Lie-algebra-valued cocycle for the circle action. Averaging a cocycle over the circle writes it as a coboundary. Integrating the resulting time-dependent element of G_c(ℝ) conjugates the circles along the path. One can do this first in the adjoint group and lift through the central isogeny; the remaining central discrepancy is locally constant. The tangent to the filtration orbit is obtained from the negative degrees in g_ c, . Griffiths transversality makes its only possible degree equal to -1 , giving t(v) and its indicated action in every representation. Higgs integrability gives the commutation relation in. The Lie algebra sits as a rational flat subbundle of 𝕍 . Normalization by the Hodge circles makes it a Hodge subvariation, and restriction of the endomorphism polarization polarizes it. Intersecting its rational fiber with the endomorphism lattice supplies an invariant full lattice. These assertions also hold for the stated rational sums of ideals. Finally, preservation of the polarization and normalization by the Hodge grading imply stability under Hodge-metric adjoint. On the real Lie algebra the negative adjoint is the Cartan involution obtained from the Weil operator. Its complex antilinear extension is the claimed compact conjugation. A compact real form of a semisimple complex algebra is the direct sum of compact real forms of its simple ideals, so this conjugation preserves each ideal.