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

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