Bribes

There's no magic formula: On the diplomacy screen, there is an "attack" option for both you and the AI (not to be confused with the "declare war" option in DA onwards). Clicking on it for the AI will bring up all the races they'd be willing to go to war with, and you can select one or more to bribe them to attack. From here on, it's conducted like a normal trade; note that the AI will never agree to attack a race with a higher military ranking.

You can also use your 'attack' option to declare war on races that AI dislikes, and can usually get a small (very small) amount of money for it.

I think an AI can be bribed to attack another that has a higher military ranking.  IIRC, I have managd that several times, including a current game (more below).  I do agree, though, that if an AI has a high mil ranking and another has a low one, that it seems impossible.

In a tight game, it can be vital to keep track of which AIs have treaties and trade routes with each other.  For example, even trade routes produce income for them and a war declaration stops that, just as a research and eco treaty being lost stops that source of resources.  Additionally, the freighters can end up coming to the player, just as long as you keep out of wars.

I have found getting and keeping all the other AIs into early wars amongst each other to be a key tactic to winning on Suicidal.  Being in other wars often makes them "too busy" to declare war against the player.

The game (DA) I am currently playing is one in which I waited too long.  The Drengin got too big earlier than I realized, decided their colony rush was over, and declared war on me.  I got several others to declare war on them (despite lower mil rankings), but it has become a hard (but fun) game.  I look to have them on the ropes now, but we have traded invasions of border worlds for a few score turns now.  They sent multiple fleets immediately at my mining bases.  I had to cede the purple, yellow and even the blue ones, but have been doggedly defending my one green one and my two red ones.  They destroyed one red one after killing my defending fleets, but I have retaken it.  Meanwhile, I have been nibbling away with invasions on their other flanks.

Also, in the meanwhile, one of the other AIs that has kept out of the war has grown and quietly began gobbling Neutrals.  I had to bribe them to peace, as the Neutrals are important to my economy.

A fun game!

Correct, did that earlier today.

Not that this is a guarantee that the bribed AI will be very succesful or aggressive in attacking... In my last game I barely escaped with a tech victory while the Terrans (of all races) were finishing off my fleet and the Korath's.

As soon as the Terrans declared war on me, I bribed the Korath into attacking them but that only bought me some time.

I've bribed the Korath to declare war on a powerful race.   All they did was sign a peace treaty with them the very next move.  This does not respect the "Ambasador is too busy to see you due to recent hostilities"  rule.  There should have been a few moves between the declaration of war and peace unless another race bribed them to agree to a peace treaty.

It's happened a few times.  The race that got bribed "kept their word" and by signing a peace treaty right away, they keep their bribe.

In one game I'm nearing the end and I do not think I get the "too busy to talk" elay any more.  Mine is the most powerful empire with the most influence and I've completed the diplo tree.  ALong the way, I built the Translators, the Restaurant and the SHowcase, or captured them.  I wonder if the delta between the empires gets large enough, if that delay is waived.  I think it is also waived between allies, so there IS a precedent in the game engine for no delays.

DO note, though, that any war, even brief, should extinguish any eco or research treaties between those two empires, and even quash the trade routes, I think.  So, there is some benefit from even a one turn war.

The delay is definitely waived between allies, but as far as I know not through any other means, no matter how high your diplomacy and influence.

A war does end treaties and trade routes, but I have always had a great deal of difficulty getting AIs with one to go to war, except in an indirect manner by bribing one to attack the other's ally (or the Altarians if they're good). It seems that the bonus for shared treaties will almost always make AIs "friendly" towards each other, which is the level of relations where they won't go to war.