# Decomposability of chain complexes

The following is stated in Luc Illusie, "Frobenius and Hodge degeneration", part 4.6.

Let $$L$$ be a bounded chain complex. There is a sequence of obstructions, first $$c_i\in \mathrm{Ext}^2(H^iL, H^{i-1}L)$$, if these vanish then secondary obstructions $$c'_i\in \mathrm{Ext}^3(H^iL, H^{i-2}L)$$ are defined, etc. $$L$$ is decomposable if and only if all these obstructions vanish.

(Decomposability here means that in the derived category it is isomorphic to a chain complex with $$0$$ differential.)

If the homology is concentrated in two subsequent dimensions, then this is proved, but in general, no proof (or construction of the obstructions) is given.

Question: Is there a reference for the general case?