The author studies the commutant and the double commutant of the algebra \(\text{alg }{\mathcal L}\) of all bounded operators on a Banach space \(X\) leaving invariant each member of a lattice \({\mathcal L}\) of subspaces of \(X\). For example, he proves that when \({\mathcal L}\) is the pentagon subspace lattice, then the only operators commuting with all the operators in \(\text{alg }{\mathcal L}\) are multiples of the identity, hence \((\text{alg }{\mathcal L})''= L(X)\). He also gives an example of an algebra \({\mathcal A}\) of operators on a complex Hilbert space which is reflexive (\({\mathcal A}=\text{alg}(\text{lat }{\mathcal A}\))), whose commutant and double commutant coincide, but are not reflexive.