A. Algorithm for BKNHC chain dynamics Splitting the BKNHC chain Hamiltonian as we obtain the corresponding measure-preserving splitting of the Liouville operator At this stage we go directly to Eqs (20). The antisymmetric Nos´e-Hoover-Bulgac- Kusnezov tensor becomes 19 H BKNHC 1 = V ( q ) (47a) H BKNHC 2 = p 2 2 m (47b) H BKNHC 3 = k$_{B}$Tζ (47c) H BKNHC 4 = k$_{B}$Tξ (47d) H BKNHC 5 = K$_{1}$ ( p$_{ζ}$ ) m$_{ζ}$ (47e) H BKNHC 6 = K$_{2}$ ( p$_{ξ}$ ) m$_{ξ}$ (47f) H BKNHC 7 = p 2 η 2 m$_{η}$ (47g) H BKNHC 8 = k$_{B}$Tη (47h) H BKNHC 9 = p 2 χ 2 m$_{χ}$ (47i) H BKNHC 10 = k$_{B}$Tχ , (47j) L$_{α}$ = B BKNHC ij ∂H BKNHC α ∂x$_{j}$ ∂ ∂x$_{i}$ . (48) ˜ B BKNHC =        0 0 0 0 0 1 0 − q 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 − 1 0 0 0 0 0 − p 0 0 0 0 − 1 0 0 0 p 0 0 − p$_{ζ}$ 0 q 0 − 1 0 0 0 0 0 0 − p$_{ξ}$ 0 0 0 − 1 0 0 p$_{ζ}$ 0 0 0 0 0 0 0 − 1 0 0 p$_{ξ}$ 0 0        , (49)