I was given a problem to compute Casimir basis functions for Lie algebra $so(4)$. I found that quadratic Casimir is $X_ik_{ij}X_j$, where $k_{ij} = Tr(X_i\cdot X_j)$ is the Killing form and $X_i$ is the matrix representation of $so(4)$.
Consider the matrix representation of $so(4)$ as $L_{ij} = E_{ij} - E_{ji}$, where $1\leqslant i<j\leqslant4$ and $E_{ij}$ is the matrix with only non-zero entry $(E_{ij})_{ij} = 1$. Then let $$X_1 = L_{12},\quad X_2 = L_{13},\quad X_3 = L_{14},\quad X_4 = L_{23},\quad X_5 = L_{24},\quad X_6 = L_{34}$$ Multiplying matrices we obtain that
$$k_{ij} = \begin{cases}0, & i\ne j \\ -2, & i=j\end{cases}$$
And so the quadratic Casimir (Enstein convention):
$$X_ik_{ij}X_j = X_ik_{ii}X_i = 2X_iX_i = 6\left(\matrix{1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1}\right)$$
My questions are:
Is it really Casimir basis function? I could not find anything about Casimir basis in any literature
Are there any other Casimir functions (or Casimir basis functions)? How to compute them?