In May 2020, Wikipedia changed their Snub Dodecahedron page, including redefining Xi as a new constant Eta. In 1985, I found an alternate solution that also uses Phi (Golden Ratio), and Eta. Here are both solutions.
$$\text{Wikipedia's Snub Dodecahedron volume =}\; \frac{12\color{Crimson}{\eta^2}\color{Black}{(3}\color{DarkGoldenrod}{\varphi} \color{Black}{+1)-}\color{Crimson}{\eta} \color{Black}{(36}\color{DarkGoldenrod}{\varphi} \color{Black}{+7)-(53}\color{DarkGoldenrod}{\varphi} \color{Black}{+6)}}{6\sqrt{(3-\color{Crimson}{\eta^2}\color{black}{)^3}}} $$
$$ \text{Alternate solution =}\; \frac{10\color{DarkGoldenrod}{\varphi}}{3}\sqrt{ \color{DarkGoldenrod}{\varphi^2} \color{Black}{+3}\color{Crimson}{\eta} \color{Black}{(}\color{DarkGoldenrod}{\varphi} \color{Black}{+}\color{Crimson}{\eta} \color{Black}{)}} \,+\,\frac{\color{DarkGoldenrod}{\varphi^2}}{2}\sqrt{ 5 + 5\sqrt{5}\color{DarkGoldenrod}{\varphi}\color{Crimson}{\eta} \color{Black}{(}\color{DarkGoldenrod}{\varphi} \color{Black}{+}\color{Crimson}{\eta} \color{Black}{)}} $$
$$ \text{where } \color{Crimson}{Eta}\text{ is defined as:} \quad \color{Crimson}{\eta\,\equiv\, \sqrt[3]{ \frac{ \color{DarkGoldenrod}{\varphi} \color{Crimson}{}}{2} + \frac{1}{2} \sqrt{ \color{DarkGoldenrod}{\varphi} \color{Crimson}{-}\frac{5}{27}}}\;+\;} \color{Crimson}{ \sqrt[3]{ \frac{ \color{DarkGoldenrod}{\varphi} \color{Crimson}{}}{2} - \frac{1}{2} \sqrt{ \color{DarkGoldenrod}{\varphi} \color{Crimson}{-}\frac{5}{27}}}} $$
Both expressions are the same number, 37.616649962733362975777..., as shown in this Wolfram Cloud Notebook. None of the Wikipedia talk notes seem to make reference to where their solution originates?