The Lorentz Boosts (for 1+1D) can be described by the Split-Complex Numbers. A Lorentz Boost in the direction of $x$ with the rapidity $\alpha$ for a 1+1D-system can be calculated using
$$q \mapsto e^{\alpha i} q.$$
Note that $i$ is the imaginary unit of the Split-Complex Numbers. The Hyperbolic Quaternions are a generalization of them, defined as $q = a + bi + cj + dk$ with $$i^2 = +1, j^2 = +1, k^2 = +1.$$
The other combinations between $i, j$ and $k$ follow the pattern of the ordinary Quaternions. Furthermore, the polar form of a Hyperbolic Quaternion uses the hyperbolic functions $\sinh$ and $\cosh$, a hyperbolic angle $\alpha$ and a (hyperbolic) Unit Quaternion $\epsilon$:
$$e^{\alpha \epsilon} = \cosh{\alpha} + \epsilon \sinh{\alpha}$$
Interestingly, and that's also part of the reason why I'm asking this question, Biquaternions (complexified Quaternions) can be used to formulate a general pure Lorentz Boost using a similar polar form (this is not the general polar form as this is only a Boost, not a general Lorentz-Transformation) which is
$$e^{\alpha h \mu} = (\cosh{\alpha} + h \mu \sinh{\alpha}),$$ using again the hyperbolic angle $\alpha$ and a Unit Quaternion $\mu$. The formula for the Lorentz Boost is the following:
$$q \mapsto (\cosh{\frac{\alpha}{2}} + h \mu \sinh{\frac{\alpha}{2}})\space q \space (\cosh{\frac{\alpha}{2}} + h \mu \sinh{\frac{\alpha}{2}}),$$
$\alpha$ being the rapidity in the direction $\mu = xi + yj + zk$ and $h$ being the imaginary unit of the complex coefficients.
Now when searching for a general Lorentz Boost formula with Hyperbolic Quaternions, I'm not able to find one and Wikipedia only states the special case formula from above with the Split-Complex Numbers. Still, the needed Biquaternions have a quite similar polar form compared to the one of the Hyperbolic Quaternions. Apart from the missing imaginary unit $h$ and the differences when simplifying $i^2, j^2$ or $k^2$, what are the reasons why there's no general Lorentz Boost formula with Hyperbolic Quaternions or if there is, how does the formula look like?
