Doubt on the consistency condition of the secondary constraint of electrodynamics

Question

On the Dirac's method for the electromagnetism, demanding consistency on the secondary constraint $X$ (which should be identically achieved since there are no further constraints),

$$ X_{\mathbf{x}}[\pi, \lambda] \equiv \int d^3x \,\lambda(\mathbf{x}) (\partial_i\pi^i-j^0)(\mathbf{x}) \approx 0 $$

I get

$$ \begin{split} 0\overset{!}{\approx }\dot{X}\approx & \{X(\mathbf{x}),H_{P}(\mathbf{y})\} =-\int d^3z \, \frac{\delta H_P(\mathbf{y})}{\delta A^{\mu}(\mathbf{z})}\frac{\delta X(\mathbf{x})}{\delta \pi^{\mu}(\mathbf{z})} \\ \approx&\int d^3z \, \frac{\delta H_P(\mathbf{y})}{\delta A^{i}(\mathbf{z})}\partial_i [\lambda(\mathbf{x}) \delta(\mathbf{x}-\mathbf{z})] \\ \therefore & \; \; 0\approx \;\partial_i \frac{\delta H_P}{\delta A^{i}}=\partial_i \partial_j F^{ji}-\partial_i j^i = -\partial_i j^i \end{split} $$

With $H_P$ being the primary hamiltonian for the electrodynamics with sources, with density

$$ \mathscr{H}_P=\frac{1}{4}F^{i j} F_{i j} -\frac{1}{2}\pi^{i}\pi_{i} -A_0 \partial_i \pi^{i} + j^{\mu}A_{\mu} + \lambda_1 \pi^0 $$

Now, the first term of $\dot{X}$ is, of course, identically zero, which guarantees the consistency of the secondary constraint on electrodynamics without sources. The problem is that I'm left with the term $\partial_i j^i$ that has no reason to be zero (differently from $\partial_{\mu} j^{\mu}=0$, of course).

It's been a while since I'm stuck here and I'd be extremely grateful if someone could at least point me at the right direction.

Edit: I started off with the lagrangian density

$$ \mathscr{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}-j^{\mu}A_{\mu} $$

Answer 1

I found my mistake a few weeks after posting the question and only now decided to post the solution here:

The silly error was just the lack of the second term in

$$ \dot{F}(x)=\{F,H\}+\frac{\partial F}{\partial t} $$

Which would then give me something like

$$ 0 \overset{!}{\approx} \partial_i \partial_jF^{ij}-\partial_ij^{i}-\partial_0j^0=-\partial_\mu j^\mu \approx 0 $$

So that, indeed, the two constraints found up to here exhaust the constraints of the theory.