What Are the Relations for the Polar Hypercomplex form $a + bi + cj + dk$? Olariu in "Complex Numbers in $N$ Dimensions" has polar hypercomplex numbers described by its generators as
\begin{gather}
\alpha^2 = \beta, \\
\beta^2 = 1, \\
\gamma^2 = \beta, \\
\alpha\beta =\beta\alpha = \gamma, \\
\alpha\gamma =\gamma\alpha = 1,\\
\beta\gamma = \gamma\beta = \alpha.
\end{gather} 
A commutative hypercomplex number can be expressed as a linear combination of real and non-real roots of $1$ and $-1$.  
My goal is to find an expression for these polar hypercomplex numbers as $a + bi + cj + dk$.  
Any help, reference would be greatly appreciated.

The answer is that it can't be done.  Because the ring is principal (as is described in answer 2) i.e. 
$a  + bi + cj + dk = A+ B \alpha +C \alpha^2 + D \alpha^3 $
there are no relations that would allow for 
$i^2 = \pm 1$, $j^2 = \pm 1$ and $k^2 = \pm 1$. 
