The Triple Riordan Group

We define the triple Riordan group, whose elements consist of $4$-tuples of power series $(g,f_1,f_2,f_3)$ with $g\in \mathbf{R}[[x^3]]$, and $f_1,f_2,f_3\in x\mathbf{R}[[x^3]]$, for an appropriate ring $\mathbf{R}$. The construction of this group generalizes that of the double Riordan group, and lays the pattern for further generalizations.