Let $X$ be a projective Fano 3fold or 4fold and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. For what $X$ is it known a semi orthogonal decomposition into indecomposable subcategories of $D^b(X)$? I guess quite a lot of them...

Idecomposability is not always easy to prove, but here is a list of Fano 3folds $X$ with $Pic(X) = {\mathbb{Z}}$ for which a decomposition with presumably indecomposable components is known: 1) index 4: $D(P^3) = < O,O(1),O(2),O(3)>$; 2) index 3: $D(Q^3) = < S,O,O(1),O(2)>$, $S$ is the spinor bundle; 3) index 2: one always have $D(X) = < B_X,O_X,O_X(1) >$; let $d = K_X^3/8$; then a) $d = 5$ then $B_X$ is generated by an exceptional pair; b) $d = 4$ then $B_X = D(C_2)$, $C_2$ is a curve of genus $2$. c) for $1 \le d \le 3$ the category $B_X$ is expected to be indecomosable. 4) index $1$, let $g_X = 1K_X^3/2$; then such decomposition is known for $g_X \in \lbrace 12,10,9,8,7\rbrace$. For those Fano 4folds which are unsections of Fano 3folds listed here a decomosition is also known. 

