The Principle of Optimality and the Bellman Equation Given (1) and (2), is it possible to show the existence of a Bellman equation (3), using Bellman's Principle of Optimality?

$$\ max \Sigma\beta^s U(C_t)$$

Subject to the following resource constraint:

$$C_t + K_{t+1}= F(K_F, E_F,S_t)$$

Where:
$$E_t = F_E(K_E,E_2,R_t, R_{t+1})$$
$$S_t = \Sigma(1-d)E_t$$
$$R_{t+1}=R_t -E_t$$
And the Bellman equation is given:

$$V_t (K_t,R_t,E_t)= max_{C_1,K_2,E_1,E_t} {U(C_t)+\beta V_{t+1}(K_{t+1},R_{t+1},E_{t+1}) + \lambda_2 (F_2 (K_2,E_t-E_1)-E_t)}$$

This problem is taken from eq. (27) and eq. (39) of the following textbook: https://are.berkeley.edu/~traeger/pdf/KarpTraegerDraft.pdf.
The specified parameters in the model are:

Consumption $C_t$
Capital $K_t$
The energy sector $E_t$
The carbon stock $S_t$
The finite resource $R_t$

Would genuinely appreciate being able to see how this is conducted on this type of model.
Recall that the Principle of Optimality states that the solution to Our Bellman Functional Equation is the same as the solution to the sequential problem if:
Assumption 1: $\Gamma(x)$ (our set of feasible values) is non-empty for all $x\in X$.
Assumption 2:$\lim_{t\rightarrow\infty}\sum_{t=0}^\infty \beta^t F(x_t,x_{t+1}) $ exists for all $\tilde{x}\in \Pi(x_0)$ ($\Pi(x_0)$ being the correspondence of $x$ considering an initial $x_0$)
Assumption 3:$|V^*(x)&lt;\infty|$ for all $x \in X$.
Assumption 4: For any $x_0$, there exists a plan $\tilde{x}\in \Pi(x_0)$ such that $u(\tilde{x})=V^*(x_0)$.
In your story here since $\beta&lt;0$ we know that this is bounded all we are doing is adding an additional constraint. This is equivalent to writing the legrangian:
$$\mathcal{L}=\sum_{t=0}^\infty\{\beta^tu(c_t)+\sum_i\lambda_{t,i}(F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}\}$$
Again I may not be iterating the whole story because there are two sectors here but this is how I'd go about thinking about if the principle of optimality is satisfied.
Hope this helps
Edit: After looking at your addendum im guessing the proceedure would be the same
$$V_t(k_t)=max_{c_t}\{u(c_t)+\lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]\}+\max_{c_{t+1},...,c_{T}}\{\sum_{t+1} \ [\beta^{t+1}u(c_{t+1})+\sum_i\lambda_{it}[F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]]\} $$
or
$$V_t(k_t)=max_{c_t}\{u(c_t)+\lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]+V(k_{t+1})\}$$