I am reading the paper https://arxiv.org/pdf/hep-th/0209056.pdf. I have a question about the SU(4) Chevalley basis.
It is well known that su(4) algebra can be written as \begin{equation} [R^i_{~j},R^k_{~l}]=\delta^k_j R^I_{~l}-\delta^i_l R^k_{~j} \end{equation} where $R^i_{~j}$ is the generator of su(4).
On the other hand we may write su(4) algebra in Chevalley basis with generator $(H_i, E^\pm_i), i=1,2,3$ \begin{equation} [H_i,H_j]=0 ,~~ [ E_i{}^{ +}, E_j{}^{-} ] = \delta_{ij} H_j ~~ \quad [H_i , E_j{}^{\pm}] = \pm K_{ji} E_j{}^{\pm} \end{equation} where $K_{ij}$ is cartan matrix for su(4) (eq.(3.25) in the paper).
Then one can relate $R_{ij}$ with Chevalley basis as follows \begin{equation} [ R^i{}_{j} ] =\left(\begin{array}{cccc} \frac{1}{4}(3H_1{+2H_2}{+H_3})& E_1{}^{+} & [E_1{}^{+},E_2{}^{+}]& [E_1{}^{+},[E_2{}^{+},E_3{}^{+}]]\\ E_1{}^{-} & \frac{1}{4}({-H_1}{+2H_2}{+H_3})& E_2{}^{+} & [E_2{}^{+},E_3{}^{+}] \\ -[E_1{}^{-},E_2{}^{-}]& E_2{}^{-} & -\frac{1}{4}(H_1{+2H_2}{-H_3}) & E_3{}^{+} \\ [E_1{}^{-},[E_2{}^{-},E_3{}^{-}]] & -[E_2{}^{-},E_3{}^{-}]& E_3{}^{-} & - \frac{1}{4}(H_1{+2H_2}{+3H_3}) \end{array}\right) \end{equation} which is (3.26) in the above paper. My question is that how to get the last equation, is there any general method?