$\bullet$ It's a lattice as it's every pair of elements has LUB and GLB.
$\bullet$ It is not a distributive lattice as-
$\Rightarrow \ b ∧ (c ∨ d) = b ∧ a = b$
$(b ∧ c) ∨ (b ∧ d) = e ∨ e = e$
$b \neq e $
$\Rightarrow$ All 3 elements- b,c and d have 2 complements.
$\Rightarrow$ It's a famous diamond structure $(M_3)$ lattice which is non-distributive. Moreover, any lattice is distributive if and only if it does not contain $M_3 \ or \ N_5$ as sublattice.
$\bullet$ A lattice is a Boolean algebra if and only if it is distributive and complemented.
It is not distributive hence not boolean algebra.