The Casimir element (Q2515178)

I found it frustrating to read this paper. The author gives an explicit derivation of what is claimed to be the fourth order Casimir operator for the Lie algebra of the orthogonal group \(\mathrm{SO}(5)\). This Casimir is known, but the author gives a proof that uses new ideas. The author correctly defines \(\mathrm{SO}(5)\) as the group of linear transformations preserving the quadratic form \(\sum_{i=1}^5 (x_i)^2\) but then considers \(\mathrm{SO}(5)\) as the group preserving the form \(x_1x_3+x_2x_4+x_3x_1+x_4x_2+x_5^2\). However this is \(\mathrm{SO}(3,2)\), not \(\mathrm{SO}(5)\)! The author introduces a basis for the Lie algebra of this group and computes the root structure explicitly. Then the author considers the adjoint representation of the Lie algebra and the characteristic polynomial of the operators in the adjoint representation. This involves computing a determinant whose matrix elements are noncommuting operators. \(\chi_B(\lambda)=\det(B-\lambda I),\quad B-\lambda I=\left(\begin{matrix} H_1-\lambda& A_1&0&A_2&-A_3\\ -A_1&H_2-\lambda &-A_2&0&-A_5\\0&A_6&-H_1-\lambda&A_4&-A_7\\ -A_6&0&-A_1&-H_2-\lambda&-A_8\\ A_7& A_8& A_3& A_5& -\lambda\end{matrix}\right).\) This is done without explicitly defining such a determinant or providing references to its properties. Here \(H_1,H_2\) form a basis for the Cartan subalgebra and \(A_1,\cdots, A_8\) form a complementary basis for the rest of the Lie algebra. The Casimir operators are identified as the coefficients of \(\lambda^k\) in the expansion of \(\chi_B(\lambda)\). The coefficients of \(\lambda^2\), \(\lambda^4\), and \(\lambda^0\) are zero and that of \(\lambda^5\) is \(-1\). The coefficient of \(\lambda^3\) is identified as the 2nd order Casimir and the coefficient of \(\lambda^4\) is the 4th order Casimir. The author proceeds to simplify these sums of products of noncommuting operators. Here another confusion occurs. Casimir operators are abstract objects in the universal enveloping algebra. However the author sometimes seems to be assuming that the Lie algebra elements are \(5\times 5\) matrices. Thus, for example, in the simplification we see the remark \(H_1H_2=0\), which is true in the \(5\times 5\) matrix realization but not for the enveloping algebra. This probably can be avoided, but certainly leads to confusion. Finally, after a long computation, a simple explicit expression for the fourth order Casimir emerges. I think that there are some interesting ideas in this paper, but the present article is a muddle.