\nu}_g$ respectively stand for the quark and gluon parts of QCD energy-momentum tensor in the Belinfante symmetrized form. By evaluating the nucleon forward matrix element of the above identities, we can prove the following important relations.

First, for the quark part, we get $$\begin{aligned}
 \hspace{10mm} L_q &\equiv& 
 \langle p \uparrow \,| \,
 M^{012}_{q-OAM} \,| \, p \uparrow \rangle \nonumber \\
 &=& \frac{1}{2} \,\int_{-1}^1 \,x \,
 [\,H^q (x,0,0) \ + E^q (x,0,0) \,] \,dx \ - \ 
 \frac{1}{2} \,\int_{-1}^1 \,\Delta q(x) \,dx , \end{aligned}$$ with $$\begin{aligned}
 M^{012}_{q-OAM} 
 \ = \ \bar{\psi} \,\left(\mbox{\boldmath $x$} \times \frac{1}{i} \,
 \mbox{\boldmath $D$} \right)^3 \,\psi
 &\neq& \left\{ \,
 \begin{array}{l}
 \bar{\psi} \,\left(\mbox{\boldmath $x$} \times \frac{1}{i} \,
 \nabla \right)^3
 \,\psi \\ 
 \bar{\psi} \,\left(\mbox{\boldmath $x$} \times \frac{1}{i} \,
 \mbox{\boldmath $D$}_{pure} \right)^3 \,\psi .
 \end{array}
 \right.\end{aligned}$$ We find that the proton matrix element of our quark OAM operator coincides with the difference between the 2nd moment of GPD $H + E$ and the 1st moment of the longitudinally polarized distribution of quarks. What should be emphasized here is that full covariant derivative appears, not a simple derivative operator nor pure-gauge covariant derivative. In other words, the quark OAM extracted from the combined analysis of GPD and polarized PDF is [*dynamical*]{} 