I wonder if somebody can provide enlightenment on my question on the polynomial below.

The polynomial has the property that if you pick up one root, the other two can be obtained by successively acting certain fractional linear transformation on it. I wonder whether this phenomenon is ubiquitous (at least to a certain family of polynomials over rational numbers, say) or sporadic.

This fact is remarked by Shanks (an excerpt is attached below), but he gives no explanation as to why that is the case nor how he obtained the result. Serre refers to this polynomial in one of his papers below, but he says nothing about the (fractional linear) group action.

In this article, Shanks says:

The cubic equation $$(a) \qquad x^3 =ax^2 + (a + 3)x + 1$$ has the discriminant $$(b) \qquad D = (a^2 + 3a + 9)^2,$$ and if $a^2 + 3a + 9$ is prime, (b) is obviously also the discriminant of the field $\mathbb{Q}(\rho)$ where $\rho$ is a root of (a). One may easily verify that the other two roots of (a) are $\rho_{2} = -l/(l+\rho)$ and $\rho_{3} = - 1/(1 +\rho_{2})$.

Serre refers to this polynomial, saying "c'est même là une extension universelle" in Théorème 3 here.