Let (L_n)_{n ≥ 0} be the Lucas sequence given by L₀ = 0, L₁ = 1, and L_n+2 = L_n+1 + L_n for n ≥ 0. In this paper, we are interested in finding all powers of two which are sums of two Lucas numbers, i.e., we study the Diophantine equation L_n + L_m = 2^a in nonnegative integers n, m, and a. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in diophantine approximation. This paper continues our previous work where we obtained a similar result for the Fibonacci numbers.