The rapid developments in orbital-angular-momentum-carrying Laguerre-Gaussian modes in recent years have facilitated progress in optical communication, micromanipulation, and quantum information. However, it is still challenging to efficiently generate bright, pure, and selectable laser modes in compact devices. We demonstrate a low-threshold solid-state laser that can directly output selected high-purity modes with high efficiency and controllability. Spin-orbital angular momentum conversion of light is used to reversibly convert the transverse modes inside a cavity and determine the output mode index. The generated and laser modes have purities of approximately 97% and approximately 93% and slope efficiencies of approximately 11% and approximately 5.1%, respectively. Our cavity design can also be easily extended to produce higher-order Laguerre-Gaussian modes and cylindrical vector beams. Such a compact laser configuration features flexible control, low threshold, and robustness, making it a practical tool for applications in superresolution imaging, high-precision interferometers, and quantum correlations.