In section 4.4.1, Boyd mentioned that to decompose the problem in ADMM, the $$A^TA$$ matrix has to be block diagonal and the objective function $$f(x)$$ has to be block separable ($$f(x) = \sum_{i=1}^N f_i(x_i)$$, where $$x_i$$ are subvectors of $$x$$). I think these conditions are quite strong, and if they are met, we can also decompose the $$x$$-update step in the classical method of multipliers. So why do we say ADMM is more decomposable than the method of multipliers?