Summary: We construct \((2n)^2\times (2n)^2\) unitary braid matrices \(\hat{R}\) for \(n\geq 2\) generalizing the class known for \(n=1\). A set of \((2n)\times (2n)\) matrices \((I,J,K,L)\) are defined. \(\hat{R}\) is expressed in terms of their tensor products (such as \(K\otimes J\)), leading to a canonical formulation for all \(n\). Complex projectors \(P_{\pm}\) provide a basis for our real, unitary \(\hat{R}\). Baxterization is obtained. Diagonalizations and block-diagonalizations are presented. The loss of braid property when \(\hat{R}\) \((n>1)\) is block-diagonalized in terms of \(\hat{R}\) \((n=1)\) is pointed out and explained. For odd dimension \((2n+1)^2\times (2n+1)^2\), a previously constructed braid matrix is complexified to obtain unitarity. \(\hat{R}\mathrm{LL}\)- and \(\hat{R}\mathrm{TT}\)-algebras, chain Hamiltonians, potentials for factorizable \(S\)-matrices, complex non-commutative spaces are all studied briefly in the context of our unitary braid matrices. Turaev construction of link invariants is formulated for our case. We conclude with comments concerning entanglements.