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III. Geometric Algebra
With the definitions of the inner product (3.1) and outer product
(3.2), the multiplicative properties of the tetrad base $\{
{{\mathbf{e}}_a}\} $ can then be summarized in the geometric
product rule
\[
{{\mathbf{e}}_a}{{\mathbf{e}}_b} = {{\mathbf{e}}_a} \cdot
{{\mathbf{e}}_b} + {{\mathbf{e}}_a} \wedge {{\mathbf{e}}_b} \]
An essential feature of the geometric product is that it is invertible,
that is, the reciprocal basis $\{ {{\mathbf{e}}^a}(x)\} $ may be
obtained algebraically:
\[ {{\mathbf{e}}^a}: =
{({{\mathbf{e}}_a})^{ - 1}}, {\quad}{{\mathbf{e}}^a}{{\mathbf{e}}_b}
= {{\mathbf{e}}^a} \cdot {{\mathbf{e}}_b} = \delta _b^a \]
This is an equally valid basis for vectors and tensors in the
geometric tangent space $G{T_x}\mathcal{M}$.
Invertibility (3.4) relies on the fact that the newly defined
geometric product (3.3) has the property of being associative,
i.e. ${\mathbf{a}}({\mathbf{bc}}) = ({\mathbf{ab}}){\mathbf{c}}$ ,
in contrast to the inner product (3.1) and the outer product (3.2),
separately. This also is key to defining more complex products such
as ${\mathbf{a}} \cdot ({\mathbf{b}} \wedge {\mathbf{c}})$ . Useful
identities valid in the GA formalism are
\[
{\mathbf{a}}({\mathbf{b}} \wedge {\mathbf{c}}) = 2({\mathbf{a}}
\cdot {\mathbf{b}}){\mathbf{c}} - 2({\mathbf{a}} \cdot {\mathbf{c}})
{\mathbf{b}}\]
\[ {\mathbf{a}} \cdot
({\mathbf{b}} \wedge {\mathbf{c}}) = ({\mathbf{a}} \cdot
{\mathbf{b}}){\mathbf{c}} - ({\mathbf{a}} \cdot
{\mathbf{c}}){\mathbf{b}} \]
\[ ({\mathbf{a}} \wedge
{\mathbf{b}}) \cdot ({\mathbf{c}} \wedge {\mathbf{d}}) =
({\mathbf{a}} \cdot {\mathbf{d}}) ({\mathbf{b}} \cdot {\mathbf{c}})
- ({\mathbf{a}} \cdot {\mathbf{c}}) ({\mathbf{b}} \cdot
{\mathbf{d}}) \]
Proofs may be found in the references below.