Atomicity of blocks in a Hilbert lattice

Question

Where can I find the proof that any block (maximal boolean subalgebra) $\mathbf{B}$ of the orthomodular lattice $\mathcal{L}$ of closed subspaces of a separable Hilbert space $\mathcal{H}$ is atomic?

Answer 1

It is not true that all blocks are atomic, e.g. in $\mathcal{H}=L^2(\mathbb{R})$ the block containing position operator is atomless. For more details see e.g. https://physics.stackexchange.com/questions/302818/complete-blocks-of-projections-and-spectral-theorem