  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
$\mathsf{C}_2$.
