Runge Kutta methods for the ODE $y'=f(y)$ are defined by the stages $k_i$, $i=1\dots s$: $$k_i=f\left(y_0+\sum_{j=1}^s a_{ij} \, k_j\right)$$ and then the time advance: $$y_1=y_0+h \sum_{j=1}^s b_i \, k_i $$ (I assume no time dependence in the ODE just to avoid clattering notation).
Some books/papers use a different notation, e.g. $g_i$, with $k_i=f(g_i)$. No problem with that (admittedly, after some confusion) and rewriting the defining equations is almost copy/paste: $$g_i= y_0+\sum_{j=1}^s a_{ij} \, f(g_j) $$ $$y_1=y_0+h \sum_{j=1}^s b_i \, f(g_i) $$ For explicit methods, choosing one set of equations is just a matter of defining different variables in the code and storing results either after or before applying the function $f$.
However, suppose we want to use implicit methods, solving the system of $n\cdot s$ algebraic equations with exact Newton's method until convergence. Each set of equations leads to a different system, so my question is: is there an established choice?, should I solve for the $k$-stages or for the $g$-stages?
I have been unable to find any hint in the literature... maybe simply no one bothers nowadays to implement fully implicit methods and alternatives such as Rosenbrock and SDIRK methods are pursued? :)