The orthorhombic boride crystal family XYB14, 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 AlLiB14 crystal. First principles, 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 AlLiB14 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 AlLiB14 is limited by the interatomic B-B bonds that span between the B layers.