  We provide a new extension of Pitt's theorem for compact operators between
quasi-Banach lattices, which permits to describe unconditional bases of finite
direct sums of Banach spaces $\mathbb{X}_{1}\oplus\dots\oplus\mathbb{X}_{n}$ as
direct sums of unconditional bases of its summands. The general splitting
principle we obtain yields, in particular, that if each $\mathbb{X}_{i}$ has a
unique unconditional basis (up to equivalence and permutation), then
$\mathbb{X}_{1}\oplus \cdots\oplus\mathbb{X}_{n}$ has a unique unconditional
basis too. Among the novel applications of our techniques to the structure of
Banach and quasi-Banach spaces we have that the space $\ell_2\oplus
\mathcal{T}^{(2)}$ has a unique unconditional basis.
