- for monks, it is very clear: no involvement. Talking about kings, ministers, armies, cities, streets etc. is improper for monks and at MN 67 the Buddha even dismisses the monks because they are engaging in those talks (tiracchanakatha) and being loud. It takes Brahma Sahampati to appear in front of him to beg him to teach again to accept teaching again.
AN 10.69 wrote:It isn't right, monks, that sons of good families, on having gone forth out of faith from home to the homeless life, should dwell engaged in various low conversations, i.e., conversation about kings, robbers, & ministers of state; armies, alarms, & battles; food & drink; clothing, furniture, garlands, & scents; relatives; vehicles; villages, towns, cities, the countryside; women & heroes; the gossip of the street & the well; tales of the dead; tales of diversity, the creation of the world & of the sea; talk of whether things exist or not.
- as for lay people, they should involve themselves in developing conviction, virtue, generosity, and wisdom, plus the instructions given at AN 8.25:
"Mahanama, inasmuch as a lay follower is possessed of faith himself, and rouses others to possess faith; is possessed of virtue himself, and rouses others to possess virtue; is possessed of liberality himself, and rouses others to possess liberality; is himself desirous of meeting with monks, and rouses others to meet with monks; is himself desirous of hearing the true Dhamma, and rouses others to hear the true Dhamma; is himself habitually mindful of the Dhamma that is heard, and rouses others to be mindful of the Dhamma; is himself ascertained of the meaning/benefit of the Dhamma that is heard, and rouses others to ascertain the meaning/benefit; having known the meaning/benefit, having known the Dhamma, is himself committed to the practice according to the Dhamma, and rouses others to be committed to the practice according to the Dhamma; in that way, Mahanama, a lay follower is engaged in his own welfare and in others' welfare."
So as far as political matters fall in either of those categories, lay people should engage in them.