Is the lattice of subspaces of a finite-dimensional scalar product space distributive?

If you look at the definition of "lattice measure" used in Popper's paper, note that he defines "finite additivity" (axiom R4) as (essentially)

For any $A$ and $B$ (not necessarily orthogonal) $$ \mu ( A \sqcup B) = \mu(A) + \mu(B) - \mu (A \sqcap B) \tag{*}$$

assuming I have understood correctly.

Look at equation (9) in Popper's paper -- the use of this form of the "finite additivity axiom", R4, appears to be essential to the argument.

However, it appears to be known (cf. e.g. equation (5) of these slides, can't find a better reference at the moment) that the above formulation of finite additivity is equivalent to the formulation

For any $A$ and $B$ disjoint/orthogonal (i.e. $A \prec B^{\perp} \iff B \prec A^{\perp}$) $$ \mu(A \sqcup B) = \mu (A) + \mu(B) \tag{**}$$

only when the lattice is distributive. In fact Popper's paper probably amounts to a proof of that fact.

I don't know which version of the finite additivity axiom Birkhoff and von Neumann used in their original paper, but mathematically speaking I guess Popper's paper can amount to saying that if Birkhoff and von Neumann had used ($*$) instead of ($**$), they were wrong because ($*$) would imply their lattice is distributive, which it is not. Along those lines, today if you look at any lecture notes that define a notion of (probability) measure on an orthocomplemented lattice, they always use ($**$) and not ($*$).