Let's consider the following number:  

$n = n\_0 + n\_1\textbf{i} + n\_2\textbf{j} + n\_3\textbf{k}$  

Where $\{1,\textbf{i},\textbf{j},\textbf{k}\}\subset\mathbb{H}$ and in turn each component can be written as following:  

$n\_i = n\_{i0} + n\_{i1}\textbf{e}\_1 + n\_{i2}\textbf{e}\_2 + n\_{i3}\textbf{e}\_2$  

Where $\{n\_{i0},n\_{i1},n\_{i2},n\_{i3}\}\subset\mathbb{R}$ and $\textbf{e}\_1$,$\textbf{e}\_2$,$\textbf{e}\_3$ form the following algebra:  

${\textbf{e}\_1}^2={\textbf{e}\_2}^2={\textbf{e}\_3}^2={\textbf{e}\_1}{\textbf{e}\_2}{\textbf{e}\_3}=+1$  

${\textbf{e}\_1}{\textbf{e}\_2}=-{\textbf{e}\_2}{\textbf{e}\_1}={\textbf{e}\_3}$  

${\textbf{e}\_2}{\textbf{e}\_3}=-{\textbf{e}\_3}{\textbf{e}\_2}={\textbf{e}\_1}$  

${\textbf{e}\_3}{\textbf{e}\_1}=-{\textbf{e}\_1}{\textbf{e}\_3}={\textbf{e}\_2}$  

Basically, n is a quaternion whose each component is hyperbolic quaternion. Alternatively it can be rewritten as hyperbolic quaternion whose each component is a regular quaternion. Do any literature deal with this construction, and if so, does this number have any special name? Quadrotessarine? Tetraquaternion? Split-sedenion?



The name of special 16-dimensional hypercomplex number
Ther are not *sedenions* because sedenions are not associative: <https://en.wikipedia.org/wiki/Sedenion>


I doubt they have a name but this algebra is well understood. The hyperbolic quaternions is just $M\_2({\mathbb R})$. You take a tensor product of them with usual quaternions ${\mathbb H}$. As a result you get $M\_2({\mathbb H})$ that deserves no special name.


