Proof of László's conjecture for \(2\times2\) matrices and extensions via the Casimir element

In 2022, Lajos László conjectured an inequality bounding the Frobenius norm of the degree-three alternating polynomial associated with three real matrices. In this paper, we prove this conjecture for arbitrary real \(2\times2\) matrices and, more generally, establish the same inequality for arbitrary complex \(2\times2\) matrices, without any normality assumption. The key idea of the proof is a geometric reformulation: the commutator behaves like the cross product, while the alternating polynomial is expressed in terms of a determinant. This approach orthogonally separates the scalar and trace-zero parts, thereby giving a structural explanation for the constant \(3/2\), its optimality, and the removal of normality assumptions. We then use representation theory and the Casimir element to prove that the inequality holds with constant \(3/n\) for every non-trivial isotypic realization of \(\mathfrak{sl}_2(\mathbb{C})\) on \({\mathbb{C}}^n\), this constant being the best one, uniformly valid over this class. As a consequence, we obtain the inequality, with the universal constant \(3/2\), for unitary representations of \(\mathfrak{su}(2)\) and for the complexifications of orthogonal representations of \(\mathfrak{so}(3)\), even in the reducible case. In particular, we obtain the optimal constant \(1\) for real skew-symmetric matrices of order three. Finally, we describe all equality cases for \(2\times2\) matrices and highlight the algebraic obstructions to a direct extension of the method to the general case of matrices of order three.