Computing the $K$-homology $K$-theory product in splitexact algebraic $KK$-theory

Explicit formulas are indicated that compute the product $z\cdot w$ of a level-one element $z\in K{K}^G(A,\mathbf{C})$ and any element $w\in K{K}^G(\mathbf{C},B)$ in splitexact algebraic $K{K}^G$-theory, or $K{K}^G$-theory for $C^{\ast }$-algebras, with very special $G$-actions. We also make such products accessible to linear-split half-exact $kk$-theory by verifying the existence of a functor from algebraic splitexact $KK$-theory to $kk$-theory.