Suppose that for some finite-dimensional vector space $V$, we have found two different decompositions:
$$V = \bigoplus_{k ∈ \mathbb Z} W_k = \bigoplus_{\ell ∈ \mathbb Z} U_\ell $$
Now assume further that at least some of these summands intersect nontrivially. Does that mean we can write
$$V = \bigoplus_{(k,\ell) ∈ \mathbb{Z}^2} W_k \cap U_\ell $$?
If not, would the fact that all the summands are actually eigenspaces of commuting linear operators with eigenvalues indexed by $k, \ell$ help?
Consider $W_1 = \langle (1,0,0) \rangle, W_2 = \langle (0,1,0)\rangle, W_3 = \langle (0,0,1) \rangle$ and $U_1 = \langle (1,1,0) \rangle, U_2 = \langle (1,-1,0) \rangle, U_3 = \langle (0,0,1) \rangle$ as subspaces of $\mathbb R^3$.