Proof of László's conjecture for \(2×2\) 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 \(2×2\) matrices and, more generally, establish the same inequality for arbitrary complex \(2×2\) 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 \(sl_2(C)\) on \({C}^n\), 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 \(su(2)\) and for the complexifications of orthogonal representations of \(so(3)\), 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 \(2×2\) matrices and highlight the algebraic obstructions to a direct extension of the method to the general case of matrices of order three.