<s>[INST] Parametrization of integral solutions of $3x^2+3y^2+z^2=t^2$ and rational solutions of $3a^2+3b^2-c^2=-1$ 1/ Is it known the parameterisation over $\mathbb{Q}^3$ of the solutions of
$3a^2+3b^2-c^2=-1$
2/   Is it known the parameterisation over $\mathbb{Z}^4$ of the solutions of
$3x^2+3y^2+z^2=t^2$
References, articles or books are welcome
Sincerely, John
 [/INST] For #1, we can take a particular solution such as $(a_0,b_0,c_0)=(0,0,1)$, and search parametric solution in the form: $(a,b,c)=(a_0+\alpha t, b_0+\beta t, c_0+t)$. Plugging it into the equation and solving for $t\ne 0$, we get:
$$t = \frac{2}{3\alpha^2 + 3\beta^2 - 1}.$$
So, we get rational parametrization with parameters $\alpha,\beta\in\mathbb Q$:
$$(a,b,c) = \bigg(\frac{2\alpha}{3\alpha^2 + 3\beta^2 - 1},\ \frac{2\beta}{3\alpha^2 + 3\beta^2 - 1},\ 1 + \frac{2}{3\alpha^2 + 3\beta^2 - 1}\bigg).$$

For #2, we can similarly parametrize $3\left(\frac{x}{t}\right)^2 + 3\left(\frac{y}{t}\right)^2 + \left(\frac{z}{t}\right)^2 = 1$ and then explicitly expand parameters as fractions, and set $t$ be the common denominator. This way we get parametrization:
$$(x,y,z,t) = \frac{p}{q}\bigg(-2uw,\ -2vw,\ 3u^2 + 3v^2 - w^2,\ 3u^2 + 3v^2 + w^2\bigg),$$
where parameters $u,v,w\in\mathbb Z$, and parameters $p,q\in\mathbb Z$ allow to scale the variables, with the requirement that $q$ represents a common divisor of the variables.
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