Suppose I have a smooth projective variety $X$, and a semi-orthogonal decomposition of its bounded derived category:

$$D^b(X)= < A, E_1, E_2, ... , E_n >$$

where the $E_i$ are fully faithful, saturated (and hence admissible) subcats of $D^b(X)$. Is A authomatically saturated/admissible? Why? If not, under what assuptions?