For a fixed positive integer $n$, the Diophantine equation
$$x^2 + y^2 + z^2 = n$$
was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the quaternion algebra $B_{-1,-1}$, characterized by the relations
$$ i^2 = -1, j^2 = -1, k^2 = (ij)^2 = -1. $$
More precisely, it is equivalent to the norm equation
$$ \operatorname{nrd}(xi + yj + zk) = n, $$
where $\operatorname{nrd}$ denotes the reduced norm of the (integral) pure quaternion $xi + yj + zk$. For me, the most interesting fact here is that for $n \not \equiv 0 \pmod 4$ the number of solutions to this equation is some multiple the Hurwitz class number $H(n)$. In the special case $n \equiv 7 \pmod 8$, it is a multiple of zero.
Now, I wonder if the same phenomenon was observed for other Diophantine equations of this kind. For example, in the quaternion algebra $B_{-1, -3}$ defined by the relations
$$ i^2 = -1, j^2 = -3, k^2 = (ij)^2 = -3, $$
we can consider the norm equation
$$ \operatorname{nrd}\left(ix + \frac{1+j}{2}y + \frac{i + k}{2}z\right) = n, $$
which is equivalent to the Diophantine equation
$$ x^2 + y^2 + z^2 + xz = n. $$
Is there a formula for the number of its solutions? (see the OEIS sequence A014453) Is it true that, for some $n$, it is equal to the multiple of the (Hurwitz?) class number of some number field, say $\mathbb Q(\sqrt{-n})$? Perhaps, a Diophantine equation
$$ x^2 + 3y^2 + 3z^2 = n $$
was studied in detail? I'd be thankful for any references.
