  The fulfillment of the space-asymptotic Poincar\'e algebra is used to derive
new higher-order-in-spin interaction Hamiltonians for binary black holes in the
Arnowitt-Deser-Misner canonical formalism almost completing the set of the
formally $1/c^4$ spin-interaction Hamiltonians involving nonlinear spin terms.
To linear order in $G$, the expressions for the $S^3p$- and the
$S^2p^2$-Hamiltonians are completed. It is also shown that there are no quartic
nonlinear $S^4$-Hamiltonians to linear order in $G$.
