Prove
Lemma 8 ([Albert columns and their pairings]). In Types I--III let F=Z(D) be…Lemma 10 ([Small derived projections]). An alternating self-paired column of…Lemma 9 ([Two circle weights]). In the irreducible tensor decomposition of a…Proposition
The action restricts to each isotypic factor: the central isotypic projectors commute with it, and a unital homomorphism from a field to each matrix factor is injective. In Type I it extends to an action of the quadratic field F⊗_ E on F^ r_B , since F is totally real. Thus r_B is even and g≤3 . The table in Lemma 8 ([Albert columns and their pairings]) gives alternating column dimensions 2,4,6 . In Type II the same dimensions follow from 2f g≤6 . Lemma 10 ([Small derived projections]) gives the full symplectic images. In Type III the initial symmetric dimensions are also 2,4,6 . Lemma 10 ([Small derived projections]) excludes two and gives the orthogonal images for four and six. Since the column dimension is g/f and gr_B≤6 , either remaining dimension forces f=r_B=1 . The injection E↪D becomes k×k↪M_2(k) ; its two nonzero complementary idempotents each have rank one. Its eigenspaces are therefore the two separate matrix copies of the column. The self-pair matches the two Hodge weights perfectly, proving the asserted equal multiplicities. For Type IV first eliminate noncommutative D . If b>1 , the divisibility fb^2 g≤6 leaves only (g,f,b, W)=(4,1,2,2). More generally, a two-dimensional column cannot have every scalar conjugate mixed. If it did, any nonzero tensor in D(W) would be a nondegenerate alternating pair tensor of type (1,1) in every conjugate. If its isomorphism is t_W:W^*→W and the polarization isomorphism is t_ opp :W^*→ W , then t_ opp t_W^ -1 :W→ W is an admissible Hodge isomorphism, since the two Hodge shifts cancel. By it belongs, after extending by the source projector, to D_k . This is impossible because D_k preserves the distinct center embeddings. If every Type IV column is pure, the circle acts by a scalar on each W_τ⊗U_τ , with opposite scalars at opposite embeddings. It therefore lies in the rational torus of center scalars. Minimality makes the Hodge group toric, so B is CM by Section. A non-CM rank-two factor must consequently have both pure and mixed embedding pairs. For f=1 there is only one pair, excluding the tuple above and also the field case f=1,a=2 . A one-dimensional column is likewise pure at every embedding and gives CM. We now have D=K . The K -algebra K⊗_ E is a quadratic field L/K exactly when E does not admit a -embedding into K ; otherwise it is K×K . The given action on Mat_ r_B (K) extends to a unital map from this K -algebra. In the field case K^ r_B is a vector space over L , so r_B is even. Hence g≤3 ; the exclusions just proved leave only f=1,g=3,r_B=2 , with quadratic K distinct from E . In the split case its two idempotents have ranks r_+,r_-≥0 summing to r_B . Thus the action may be noncentral, as in a(a, a) on K^2 when K=E . Diagonalizing the idempotents by a K -matrix only changes how the matrix copies in each opposite pair of K -columns are assigned to the two E -eigenspaces; these matrix identifications are admissible. In the split case fa=g≤6 , a≥2 , and the exclusion of f=1,a=2 give exactly the displayed list. When f=1 the column is mixed and Lemma 10 ([Small derived projections]) applies. When f>1 its dimension is two or three. Its restriction to R_B is irreducible by the observation following ; since its dimension is greater than one, that action is nontrivial. Lemma 9 ([Two circle weights]) and the displayed small table force SL_2 or SL_3 , since two nontrivial factors already have dimension at least four. This holds also at a pure column: the active circle differential may be at another scalar-conjugate column. The rank-two pure/mixed assertion was proved above. Finally, let ρ_B be the restriction of Hdg(H^1(A)) to H_B . Its rational algebraic image contains the circle of B , hence contains J_B by minimality. Conversely, ρ_B^ -1 (J_B) is a rational algebraic subgroup of Hdg(H^1(A)) containing the circle of A . Minimality of Hdg(H^1(A)) forces this inverse image to be the whole group. Thus the image is J_B , and the derived projection is also surjective. The images just computed are therefore the projections of the actual group of A .