We describe an all-electron implementation of the Bethe-Salpeter equation (BSE) for the calculation of optical absorption spectra in the full-potential linearized augmented-plane-wave (FLAPW) method. So far, FLAPW implementations have resorted to a simple plane-wave basis for the bare and screened Coulomb potentials, thereby forgoing the all-electron description to some extent. In contrast, we expand the interaction potentials in the all-electron mixed basis. The Coulomb divergence is treated analytically, allowing for anisotropic screening. As in most implementations, the BSE is solved by the diagonalization of a two-particle Hamiltonian matrix, whose dimension is proportional to the number of $𝐤$ points. Due to the large number of $𝐤$ points required to converge the BSE, the resulting matrix becomes large even for small unit cells. We describe a method that exploits the crystal symmetries to accelerate the construction and diagonalization of the two-particle Hamiltonian. In particular, we employ group theoretical tools to bring the Hamiltonian into block-diagonal form. Furthermore, it is shown that often only one of the blocks needs to be taken into account for the optical absorption spectrum leading to a considerable speedup of the diagonalization step. The code allows for the inclusion of spin-orbit coupling and is parallelized with the possibility of storing the Hamiltonian in distributed memory over many nodes, keeping the memory demand on a single node low. To validate our implementation, we show optical absorption spectra and report exciton binding energies for bulk Si, LiF, and MoS${}_2$. By exploiting the crystal symmetries, we can reduce the dimension of the Hamiltonian matrix of Si by a factor of five, resulting in a theoretical 125-fold speedup for the leading cubic scaling term of the diagonalization. The calculated spectra agree with available theoretical and experimental spectra from the literature. The calculated exciton binding energies of 22~meV and 76~meV for Si and MoS${}_2$ are closer to experimental values than in previous BSE studies.