Lemma 12 ([Independence of the small columns]). In Lemma 11 ([The small column list]), the listed symplectic, orthogonal, and special linear derived images are independent, after identifying matrix copies and pairing opposite CM …

Split each so_4 target into its two simple ideals. The Lie image of the derived group of A is a semisimple subdirect algebra of the resulting product of simple target algebras. A projection of a simple source ideal onto a simple target is either zero or an isomorphism; exactly one source ideal maps nontrivially to each target. Indeed its image is an ideal, and the images of different source ideals commute. Thus failure of independence is exactly a graph link identifying two isomorphic target ideals. The link relation is invariant under scalar Galois transport. First consider columns mixed after every scalar conjugation, apart from the two factors of an SO_4 target and the so_6_4 case treated next. An isomorphism of the target Lie algebras in our list identifies their standard modules up to duality in the special linear case. Replace one column by its opposite in the dual case. The intertwiner equations are defined over k ; choose a nonzero derived intertwiner t over k . Its space is a line by irreducibility, and normality of the derived group makes this line stable under the full Hodge group. Writing h(z) for the Hodge-circle action, we have h(z)t h(z)^ -1 =z^m t for an integer m . Because t is an isomorphism, the target weight spectrum is the source spectrum translated by m . Both sets are -1,1 , so m=0 . The same argument applies to every scalar conjugate of t . Extending it by its source projector and using puts it in End_ Hdg (H^1(A,))⊗k . This commutant may mix only matrix copies within one central embedding and one simple isogeny type. The putative link contradicts the choice of different columns. Self-pairs force mixedness in every conjugate, and a non-CM quadratic-center column is mixed in both conjugates. Thus this argument covers all ordinary links outside the partially pure cases. The only unequal standard representations of isomorphic simple targets in this range, outside SO_4 , arise from so_6_4 . A Type III SO_6 factor has dimension six and exhausts A , so no other non-CM factor can provide such a link. Consider next a Type IV factor with both pure and mixed columns. We first exclude links among its own columns. Here r_B=1 , and the columns over one embedding of E are the f=2 or 3 embeddings of K/E . The action of Gal(k/E) on them is transitive. An invariant equivalence relation on a prime-sized transitive set is either discrete or universal, since its equivalence classes have equal size. On a column the derived part of the circle differential is its traceless part: the connected center acts by scalars. It vanishes on a pure column and is nonzero on a mixed column. A graph link identifies the two projections of the same circle differential, so it preserves whether that differential is zero. The universal relation would link a pure column to a mixed one and is impossible. Thus all ideals within this factor are independent. We must also exclude links to another isogeny factor. Only the quartic-center fourfold can have an external non-CM partner; the other partially pure cases have dimension six. Its two A_1 ideals are interchanged by Galois. The remainder has dimension at most two and inherits E . By Lemma 11 ([The small column list]), a non-CM A_1 target there can only come from a non-CM elliptic curve repeated twice or from a Type II surface. In either case there is only one absolute A_1 target and it is Galois-stable. Linking it to one member of the interchanged pair would link it to the other as well, contradicting their independence. It remains to treat SO_4 . Its column is k^2⊗k^2 for its two A_1 ideals. A nonzero semisimple traceless differential on k^2 has spectrum a,-a , a≠0 . If both factor differentials were nonzero, the tensor spectrum ±a±b would have at least three values. Hence at most one factor differential is nonzero. The self-pair makes the column mixed, so at least one is nonzero. These are the two absolute ideals of this rational factor, since its action on its sole column, repeated twice, is faithful. If both ideals were separately Galois-stable, the one with zero differential would descend to a rational adjoint quotient on which the circle acts trivially, contrary to minimality. Hence Galois interchanges them. The same remainder argument as in the preceding paragraph excludes an external link to either one. All graph links have been excluded. The Lie image is therefore the full product. A connected closed subgroup of the product of the connected classical groups with this full Lie algebra is the product itself. This proves the required independent derived image; no independence of central characters has been asserted.