An explicit product for the on-shell higher-spin algebra in $Ad{S}_d$-space

For $d>3$ we construct the product of the bosonic on-shell higher-spin algebra directly on the AdS-traceless representatives. We find the traceless projector in closed form, determine all linear relations among the trace-producing operators, and prove that every class modulo the $\mathfrak{s}\mathfrak{p}(2)$ ideal has a unique traceless representative. The resulting product is an explicit differential operator, which we test on the $\mathfrak{s}\mathfrak{o}(d-1,2)$ generators. A toy model shows how the factorisation lifts from bilinear products to trilinear ones in minimal $A_{\mathrm{\infty }}$-algebras.