The split-complex numbers are a special case of Clifford Algebra. This can be seen as follows. Let 
. Let 
with 
be a basis element of 
. Then the Clifford Algebra 
is defined as follows. The elements of the Clifford algebra are 
. Setting the rule for the Clifford product 
as
(1) 
yields
![\[(\alpha_1 + \beta_1 e_1) \vee (\alpha_2 + \beta_2 e_1) = (\alpha_1 \alpha_2 + \beta_1 \beta_2) + (\alpha_1 \beta_2 + \beta_1 \alpha_2) e_1\]](https://konrad.burnik.org/wordpress/wp-content/ql-cache/quicklatex.com-4ce7a9a11f1027330167c86c77fcc3a4_l3.png "Rendered by QuickLaTeX.com")
The isomorphism is
, 
. On the other hand, setting the rule to 
or 
in (1) yields a correspondence with dual-numbers or complex numbers respectively. I am planning to expand on this fact in one of the subsequent posts.
