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We compute the group of $K_1$-zero-cycles on the second generalizedinvolution variety for an algebra of degree 4 with symplectic involution. Thisdescription is given in terms of the group of multipliers of similitudesassociated to the algebra with involution. Our method utilizes the framework ofChernousov and Merkurjev for computing $K_1$-zero-cycles in terms of$R$-equivalence classes of prescribed algebraic groups. This gives acomputation of $K_1$-zero-cycles for some homogeneous varieties of type$\mathsf{C}_2$.

Zero-cycles with coefficients for the second generalized symplectic involution variety of an algebra of degree 4