László's three-matrix inequality for two-by-two matrices and Casimir extensions

László conjectured an upper bound for the Frobenius norm of the degree-three standard polynomial evaluated at three real matrices. We prove the conjecture for arbitrary real two-by-two matrices and, more generally, establish the same inequality for arbitrary complex two-by-two matrices, without any normality assumption. The proof uses a coordinate model of the traceless subspace in which the commutator is represented by a modified cross product and the alternating polynomial by a determinant. This yields the sharp constant \(3/2\) and a complete classification of the equality cases. We also identify a scalar Casimir criterion for three-dimensional Lie subalgebras isomorphic to \(\mathfrak{sl}_2(\mathbb{C})\). It gives the constant \(3/n\) whenever the Casimir operator is scalar on \(\mathbb{C}^n\), in particular for irreducible and isotypic realizations. This constant is sharp on the isotypic class. As consequences, the inequality holds with constant \(3/2\) for unitary \(\mathfrak{su}(2)\)-representations and complexified orthogonal \(\mathfrak{so}(3)\)-representations, and with the sharp constant \(1\) for real skew-symmetric three-by-three matrices.