We prove Lieb–Thirring inequalities with optimal, semiclassical constant in higher dimensions by following the Laptev–Weidl approach of "lifting in dimension." We introduce Schrödinger operators with matrix-valued potentials and show how Lieb–Thirring inequalities with semiclassical constants for such operators in one dimension imply the Lieb–Thirring inequality with semiclassical constant in higher dimensions. Subsequently, we prove a sharp Lieb–Thirring inequality in one dimension with exponent 3/2 for Schrödinger operators with matrix-valued potentials. We give a complete proof using the commutation method by Benguria and Loss. We also sketch the original proof by Laptev and Weidl based on trace formula for Schrödinger operators with matrix-valued potentials.