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I think you're mentioning solubility rules so you don't include groups 1 and 2 in the reaction because they don't affect pH (totally dissolve in water and are present on both sides of the reaction so you can cancel them out). Example is NaOH and also salts like NaCl don't affect pH. Conjugate bases/acids though of relevant molecules will affect pH.
The general rule is that the stronger an acid/base is, the weaker its conjugate. The weaker an acid/base is, the stronger its conjugate. Group 1/2 cations (with the exception to Magnesium, but that's a topic for a more advanced chemistry class) do not affect pH because they are mostly stable and those ions are not strong enough to break the O-H bonds in water to act as a base. However, transition metal cations that are highly charged (eg. +3) are very polarizing and can affect pH.
Group 1 and group 2 cations are typically cations from a strong acid or base. For example the Na+ of NaOH and K+ from KOH. If you look at the reaction for both of these in the presence of water you would see that since these are strong bases they full disassociate. The cation thus has no point or ability to participate in a reverse reaction and is basically an extremely weak conjugate acid that cannot contribute to the pH. Like the others said it is also a spectator ion on both sides of the reaction so it doesn't change the concentration.
Conjugate acids or conjugate bases of strong acids are very weak. This is due to the fact that Kw=Ka*Kb. Since Kw is a constant, when one of the K increases the other must decrease. So take for example HCl, it will have a very high Ka value since it dissociates completely which means the Kb will be a very small value which represents the conjugate base of Cl-.
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