This is a non-standard usage of the term "basis" (see my response to @DrChinese just now).the concept of entanglement admits a basis dependence
Again, this is a non-standard usage of the term "basis", and this non-standard usage is confusing you (and it might be confusing @DrChinese as well). The operators I wrote down in #152 manifestly commute; you can compute it explicitly if you want, it's straightforward. But those operators acts on different degrees of freedom in the Hilbert space: the ##A## operators act on the "Alice's particle spin" degree of freedom (i.e., on Alice's qubit), while the ##B## operators act on the "Bob's particle spin" degree of freedom (i.e., on Bob's qubit).in the entangled basis by definition the operators won't commute (they describe causally distinguished particle events in an inseparable state and the HUP applies) and that's why the ones you wrote are not valid in the entangled basis
In the language of the paper you linked to, these are measurements on different, disjoint, orthogonal subspaces of the Hilbert space, and with respect to each of these measurements, there is no "entanglement" in the sense the paper is using the term. Note that the definition of "entanglement" used in this paper is also non-standard, which is certainly going to be confusing when you try to relate what this paper is saying to the rest of the extensive literature on this topic, since the overall two-qubit state I described in post #152 is certainly entangled by the standard definition, yet by this paper's definition, the measurements on it that I described are not.