electromagnetism - Canonical Stress Tensor for the Free Electromagnetic Field

I have the followwing Lagrangian for the free electromagnetic field,

$$\mathcal{L} = -\frac{1}{4} F^{\mu \nu}F_{\mu \nu},$$

and the canonical stress tensor is,

$$T^{\alpha \beta}=\frac{\partial \mathcal{L}}{\partial \left(\partial_{\alpha} A^{\lambda}\right)}\partial^{\beta} A^{\lambda}-g^{\alpha \beta}\mathcal{L}.$$

My $T^{0i}$ is, $T^{0i} = (\vec{E}\times \vec{B})_{i}$

but it's the symmetric stress tensor, and I need to find it:

$$T^{0i} = (\vec{E}\times \vec{B})_{i} + \nabla \cdot (A_{i}\vec{E})$$

I don't understand where this divergence term came from. How can I do this asymmetric $T^{0i}$ tensor?

(Just for reference, see Jackson - Classical electrodynamics, Third ed.- page 606)