proving $\Box(p \to q) \to (\Box p \to \Box q)$ or $(\Box p \land \Box q) \to \Box (p \land q)$ from necessitation and other propositions

Does anyone know of any propositions that would suffice, along with the necessitation rule, to prove either of the following two propositions?

$\Box(p \to q) \to (\Box p \to \Box q)$

$(\Box p \land \Box q) \to \Box (p \land q)$

I know the first is usually taken as axiomatic in modal logic, but I'm trying to find other propositions that, together with necessitation (that all theorems of propositional logic are necessary truths), can prove the two propositions above.

Obviously you can trivially prove the distribution axiom ($\Box(p \to q) \to (\Box p \to \Box q)$) in the traditional C.I Lewis's axiomatic system K, but not from the necessitation with other propositions, otherwise it's not required to have this as an axiom.

Of course you can use a natural deduction system K to prove both of your above propositions easily, even the necessitation rule itself can be derived from such a ND system which usually has the $\Box$E and $\Box$ In modal rules in addition to the usual PL rules as referenced here:

Let us take as an example the ND formalization...; for simplicity we restrict considerations to rules for $\Box$ (necessity). ($\Box$ E) is obvious: $\Box \varphi \vdash \varphi$. With ($\Box$ I) the situation is more complicated since it is based on the following principle:

If $\varphi_1, ..., \varphi_n \vdash \psi$, then $\Box\varphi_1, ..., \Box\varphi_n \vdash \Box\psi$

Right after this quoted section you'll see a typical ND proof of your first proposition without any axioms or even the necessitation involved.

To prove your 2nd proposition under such system is also easily derived as a single step per above $\Box$ I rule:

$$\frac{p, q \vdash p \land q}{\Box p, \Box q \vdash \Box (p \land q)}$$

Or alternatively you can use a Fitch style subproof similar to the reference proving your 1st proposition...