Generating Geometrical Elements for Any Space-Time

To distinguish time from space, use real Clifford algebras $\bf {R}$$_{n;s}$, where $n$ is the number of dimensions and $s$ is their signature ($s=-n, -n+2, {\ldots}$, or $n$). $\bf{R}$$_ {n;s}$ is isomorphic to algebras of real, complex, or quaternionic matrices ${\rm {\bf R}}(2^{\textstyle{n \over {\rm {\bf 2}}}})$, ${\rm {\bf C}}(2^{\textstyle{{n-1} \over 2}})$, or ${\rm {\bf H}}(2^{\textstyle{{n-2} \over 2}})$, or of block diagonal matrices ${ }^2{\rm {\bf R}}(2^{\textstyle{{n-1} \over 2}})\mbox{ }$ or ${ }^2{\rm {\bf H}}(2^{\textstyle{{n-3} \over 2}})\mbox{ }$, for $\vert (s~$+~3)$_{mod8}$~-~4$\vert $ = 1, 2, 3, 0, or 4, respectively. Each of the $n$ basis vectors $\bf{e} $$_{\nu}$ satisfies ${\rm {\bf e}}_\mu {\bf{\cdot}}~ {\rm {\bf e}}_\nu =\eta _{\mu \nu } {\rm {\bf I}}_{n;s} $, where the $\bf {e}$$_{\nu}$ are orthogonal $\eta_{\mu \nu }=0$ for $\mu \ne \nu $ and normalized $\eta_{\mu \nu }=+1$ for $p$ space-like dimensions and $\eta_{\mu \nu }=-1$ for $q$ time-like dimensions) and where $\bf{I}_{n;s}$ is the identity matrix whose rank is given by the isomorphisms above. The geometrical elements are the scalar $\bf{I}_{n;s}$, basis vectors $\bf{e}_ {\nu }$, and their products (bivectors, trivectors, etc.) up to the pseudo-scalar $n$-volume $\bf{J}_{n;s}~=~\bf{e}_{0}~\bf{e}_ {1}~\bf{\cdot ~\cdot ~\cdot ~e}_{n-1}$. Now $\left( {{\rm {\bf J}}_{n;s} } \right)^2=(-1)^{\textstyle{{s(s-1)} \over 2}}{\rm {\bf I}}_{n;s} =\sigma _s {\rm {\bf I}}_{n;s} $. The direct product of $\bf{R}$$_{n;s}$, with $n$ orthonormal basis vectors $\bf{e}$$_{\nu }$ with signature $s$, and $\bf{R}$$_{n';s'}$, with $n'$ orthonormal basis vectors $\bf{e}$$_{\nu '}$ with signature $s'$, is ${\rm {\bf R}}_{n+n';s+s'\sigma _s } $, with $n+n'$ orthonormal basis vectors { ${\bf{e}}_{\nu }\otimes \bf{I} _{n';s'}, \bf{J}_{n;s}\otimes \bf{e}$$_{\nu '}$ } with signature ${s+s'\sigma _{s}}$, for even positive $n$. Orthonormal basis vectors for any positive $n$ with any possible signature can be generated from the two orthonormal basis vectors of the Minkowskian plane.