3-class field towers with 2 or 3 stages

For quadratic fields $k=\mathbb{Q}(\sqrt{d})$ with discriminant $d$, $3$-class group ${\mathrm{C}\mathrm{l}}_3(k)\simeq (\mathbb{Z}\mathrm{/}3\mathbb{Z}{)}^2$, and four $\mathit{\text{simple}}$ $3$-principalization types $\mathrm{\varkappa }(k)\in \{(1122),(3122),(1231),(2231)\}$, we establish necessary and sufficient conditions for the Galois group $S=\mathrm{G}\mathrm{a}\mathrm{l}({\mathrm{F}}_3^{\mathrm{\infty }}(k)\mathrm{/}k)$ of the unramified Hilbert $3$-class field tower of $k$ to coincide with the Galois group $M=\mathrm{G}\mathrm{a}\mathrm{l}({\mathrm{F}}_3^2(k)\mathrm{/}k)$ of the maximal metabelian unramified $3$-extension of $k$. In the case of non-coincidence, we study the path between $M$ and $S$ in the descendant tree of the elementary bicyclic $3$-group $(\mathbb{Z}\mathrm{/}3\mathbb{Z}{)}^2$. For two $\mathit{\text{complex}}$ $3$-principalization types $\mathrm{\varkappa }(k)\in \{(2122),(4231)\}$, we show that infinitely many non-metabelian possible Galois groups $S=\mathrm{G}\mathrm{a}\mathrm{l}({\mathrm{F}}_3^{\mathrm{\infty }}(k)\mathrm{/}k)$ with presumably unbounded derived length $\mathrm{d}\mathrm{l}(S)$ share a common metabelianization $M=S\mathrm{/}{S}^{\mathrm{\prime }\mathrm{\prime }}$, whence only partial criteria can be stated. Minimal discriminants $d>0$ with assigned simple $3$-principalization type $\mathrm{\varkappa }(k)$ and fixed length ${\mathrm{\ell }}_3(k)\in \{2,3\}$ of the $3$-class field tower are determined experimentally for nilpotency class $\mathrm{c}\mathrm{l}(M)\in \{5,7,9,11\}$ under assumption of the generalized Riemann hypothesis.