Maximal hypercubes in Fibonacci and Lucas cubes

The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1's. The Lucas cube $\Lambda_n$ is obtained from $\Gamma_n$ by removing vertices that start and end with 1. We characterize maximal induced hypercubes in $\Gamma_n$ and $\Lambda_n$ and deduce for any $p\leq n$ the number of maximal $p$-dimensional hypercubes in these graphs