I'm reading this paper by Dong where he proposed a general formula to obtain entanglement entropy (EE) for a given gravitational action. The EE can be obtain by:

\(S_{EE}=2\pi\int d^{d}y\sqrt{g}\Bigg\{\frac{\partial L}{\partial R_{z\bar zz\bar z}}+\sum_{\alpha}\Big(\frac{\partial^{2}L}{\partial R_{zizj}\partial R_{\bar zk\bar zl}}\Big)_{\alpha}\frac{8K_{zij}K_{\bar zkl}}{q_\alpha+1}\Bigg\}\)

Now suppose we have the following Lagrangian:

\[L=\lambda_1 R^{2}+\lambda_2 R_{\mu\nu}R^{\mu\nu}+\lambda_3 R_{\mu\rho\nu\sigma}R^{\mu\rho\nu\sigma}\]

then we get,

\(S_{EE}=-4\pi\int d^{d}y\sqrt{g}\Bigg[2\lambda_1 R+\lambda_2\Big(R^{a}_{\ a}-\frac{1}{2}K_a K^{a}\Big)+2\lambda_{3}\Big(R^{ab}_{\ ab}-K_{aij}K^{aij}\Big)\Bigg]\)

where the terms with extrinsic curvature are coming from the second order differential in first equation. He calls this terms the Anomaly term. I wonder if anyone can explain or show me explicitly how to obtain the extrinsic curvature terms just like above. I did the differentiation but when I do the contractions all appropriately I don't get the nice form above.