Exponentials of Cliffors: Hyperbolic, Null, and Circular Functions
The exponential of a cliffor 
is defined as
If we assume that 
is a scalar, then by the law of trichotomy, one and only one of the following cases is true:
.
Here,
and
Let 
be another spatial cliffor. If we decompose as
,
where 
and 
are the components of that commute and anticommute with ,
,
.
From the definition of the exponential of in Eqs. (2a) to (2c), we can easily see that
,
.
That is, only the component of that anticommutes with flips the sign of 
; the component that commutes with leaves the exponential unchaged after a reordering of the factors.
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