  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.
