There are lots of ways to start wars in a democracy. Of all the forms of government, my favorite one to fight a huge war with lots of units, even bloodlust, is democracy.
Besides ideas suggested so far, one of the easiest ways to start a way is with a caravan/freight. Move it into the enemy territory, block their RRs, occupy their fortresses. The AI does not like that, and it can't get rid of your freight!
The AI really has a cow when you park a spy right next to their city. Double it up so it won't be expelled. The sneak attack is "in the mail..."
Placing inferior unit next to their good units. If you have an obsolete offensive unit, like a Legion, put it next to a rifleman (outside their city radius). Especially if it blocks his road/RR. It probably won't last long
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Move a spy up to an AI city and demand they withdraw their troops. That bothers them sometimes!
Get the hawks on your side. Get stabbed in the back several times, and the hawks will eventaully rescue you and give you your desires, even without the UN.
So I was wodering if there's a way to fight without declare war on the AI.
Yes, there is ... it is called subversion, and all you need is a dip/spy and some (a lot) of gold!