3-class field towers with 2 or 3 stages

For quadratic fields $k=\mathbb{Q}(\sqrt{d})$ with discriminant $d$, $3$-class group $\mathrm{Cl}_3(k)\simeq (\mathbb{Z}/3\mathbb{Z})^2$, and four \textit{simple} $3$-principalization types $\varkappa(k)\in\lbrace (1122),(3122),(1231),(2231)\rbrace$, we establish necessary and sufficient conditions for the Galois group $S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)$ of the unramified Hilbert $3$-class field tower of $k$ to coincide with the Galois group $M=\mathrm{Gal}(\mathrm{F}_3^2(k)/k)$ 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 $(\mathbb{Z}/3\mathbb{Z})^2$. For two \textit{complex} $3$-principalization types $\varkappa(k)\in\lbrace (2122),(4231)\rbrace$, we show that infinitely many non-metabelian possible Galois groups $S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)$ with presumably unbounded derived length $\mathrm{dl}(S)$ share a common metabelianization $M=S/S^{\prime\prime}$, whence only partial criteria can be stated. Minimal discriminants $d>0$ with assigned simple $3$-principalization type $\varkappa(k)$ and fixed length $\ell_3(k)\in\lbrace 2,3\rbrace$ of the $3$-class field tower are determined experimentally for nilpotency class $\mathrm{cl}(M)\in\lbrace 5,7,9,11\rbrace$ under assumption of the generalized Riemann hypothesis.