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This paper is devoted to constructing "new" admissible subcategories andsemi-orthogonal decompositions of triangulated categories out of "old" ones.For two triangulated subcategories $T$ and $T'$ of a certain $D$ and adecomposition $(L,R)$ of $T$ we look either for a decomposition $(L',R')$ of$T'$ such that there are no non-zero $D$-morphisms from $L$ into $L'$ and from$R$ into $R'$, or for a decomposition $(L_D,R_D)$ of $D$ such that $L_D\capT=L$ and $R_D\cap T=R$. We prove some general existence statements (that alsoextend to semi-orthogonal decompositions with any number of components) andapply them to various derived categories of coherent sheaves over a scheme $X$that is proper over a noetherian ring $R$. This gives a one-to-onecorrespondence between semi-orthogonal decompositions of $D_{perf}(X)$ and$D^b_{coh}(X)$; the latter extend to $D^-_{coh}(X)$, $D^+_{coh}({Qcoh}(X))$,$D_{coh}({Qcoh}(X))$, and $D({Qcoh}(X))$ under very mild conditions. Inparticular, we obtain a vast generalization of a theorem of J. Karmazyn, A.Kuznetsov, and E. Shinder.  These applications rely on recent results of Neeman that express$D^b_{coh}(X)$ and $D^-_{coh}(X)$ in terms of $D_{perf}(X)$ along with its newvariations corresponding to $D^+_{coh}({Qcoh}(X))$ and $D_{coh}({Qcoh}(X))$. Wealso discuss an application of this theorem to the construction of certainadjoint functors.

Producing "new" semi-orthogonal decompositions in arithmetic geometry