  The orthorhombic boride crystal family XYB$_{14}$, where X and Y are metal
atoms, plays a critical role in a unique class of superhard compounds, yet
there have been no studies aimed at understanding the origin of the mechanical
strength of this compound. We present here the results from a comprehensive
investigation into the fracture strength of the archetypal AlLiB$_{14}$
crystal. First-principles, \textit{ab initio}, methods are used to determine
the ideal brittle cleavage strength for several high-symmetry orientations. The
elastic tensor and the orientation-dependent Young's modulus are calculated.
From these results the lower bound fracture strength of AlLiB$_{14}$ is
predicted to be between 29 and 31 GPa, which is near the measured hardness
reported in the literature. These results indicate that the intrinsic strength
of AlLiB$_{14}$ is limited by the interatomic B--B bonds that span between the
B layers.
