Operator splitting method on three assets black scholes equation

Currently I am studying finite difference method on derivatives with three (or more) underlyings and little bit confused on operator splitting method because two papers have different result.

For the follwing differential equation,
$$
u_tau(x,y,z,tau) = rxu_x+ryu_y+rzu_z+frac{1}{2}sigma_x^2x^2u_{xx}+frac{1}{2}sigma_y^2y^2u_{yy}+frac{1}{2}sigma_z^2z^2u_{zz}+rho_{xy}sigma_xsigma_yxyu_{xy}+rhosigma_ysigma_zyzu_{yz}+rho_{zx}sigma_zsigma_xu_{zx}-ru
$$

First paper, Comparision of numerical schemes on multi-dimensional black-scholes equations ended up with

$$
frac{partial u}{partial tau}=L_xu+L_yu+L_zu
$$
where
$$
L_xu = frac{1}{2}sigma_x^2x^2u_{xx}+rxu_x+frac{1}{2}rho_{xy}sigma_xsigma_yxyu_{xy}+frac{1}{2}rho_{xz}sigma_zsigma_xu_{xz}-frac{1}{3}ru \
L_yu = frac{1}{2}sigma_y^2y^2u_{yy}+ryu_y+frac{1}{2}rho_{yx}sigma_ysigma_xyxu_{yx}+frac{1}{2}rho_{yz}sigma_ysigma_zu_{yz}-frac{1}{3}ru \
L_zu = frac{1}{2}sigma_z^2z^2u_{zz}+rzu_z+frac{1}{2}rho_{zx}sigma_zsigma_xzxu_{zx}+frac{1}{2}rho_{zy}sigma_zsigma_yu_{zy}-frac{1}{3}ru
$$
(Original paper took derivative wrt $t$ and I changed to $tau = T-t$ here)

and the second paper, A practical finite difference method for the three-dimensional black-scholes equation defined them as
$$
L_xu = frac{1}{2}sigma_x^2x^2u_{xx}+rxu_x+frac{1}{3}rho_{xy}sigma_xsigma_yxyu_{xy}+frac{1}{3}rho_{yz}sigma_ysigma_zyzu_{yz}+frac{1}{3}rho_{xz}sigma_xsigma_zxzu_{xz}-frac{1}{3}ru \
L_yu = frac{1}{2}sigma_y^2y^2u_{yy}+ryu_y+frac{1}{3}rho_{yx}sigma_ysigma_xyxu_{yx}+frac{1}{3}rho_{yz}sigma_ysigma_zyzu_{yz}+frac{1}{3}rho_{xz}sigma_xsigma_zxzu_{xz}-frac{1}{3}ru \
L_zu = frac{1}{2}sigma_z^2z^2u_{zz}+rzu_z+frac{1}{3}rho_{xy}sigma_xsigma_yxyu_{xy}+frac{1}{3}rho_{zy}sigma_zsigma_yzyu_{zy}+frac{1}{3}rho_{zx}sigma_zsigma_xzxu_{zx}-frac{1}{3}ru \
$$

So I wonder which derivation I should use or they are just indifferent