A^2=5 I (where I is the 6x6 identity matrix). This implies that A has eigenvalues -\sqrt{5} and \sqrt{5}, both with multiplicity 3 since A is symmetric and of trace zero. The matrix A+\sqrt{5}I induces thus an Euclidean structure on the quotient space \mathbb R^6/\ker(A+\sqrt{5}I) which is isomorphic to \mathbb R^3 since