The orthorhombic boride crystal family $XY{\mathrm{B}}_{14}$, 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 ${\mathrm{AlLiB}}_{14}$ 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 ${\mathrm{AlLiB}}_{14}$ 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 ${\mathrm{AlLiB}}_{14}$ is limited by the interatomic B-B bonds that span between the B layers.