I'm reading "Introduction to Classical Integrable Systems". In pp. 14 the proposition of r-matrices there's fomula like $$\{L_1, L_2\} = [r_{12}, L_1] − [r_{21}, L_2]$$.
If my understand is correct, $L$ is a matrix (function on phase space), $L_{1},\,L_{2}$ and $r_{12},\,r_{21}$ are tensors of matrices, so how did Lie bracket of two tensor defined? any helps are appreciated