It's been long enough since FO3 for him to be dead and buried.
...do we actually know when it takes place
We know that FO3 began exactly 200 years after the bombs fell. Elder Lyons was already old as fuck and preparing to die at that point. I don't think we've heard anything definitive about FO4 beyond "roughly 200 years later", but I'd guess that it was enough difference (which could mean as few as 1-2 years, honestly) for Lyons to be out of the picture. That's if you want to go with the interpretation that the D.C. BoS are involved, rather than just the Midwesterners.
Also, because I really don't see Lyons deciding to start a crusade against the Institute when the Wasteland around D.C. is still fucked up and full of assorted minor assholes. The fact that the BoS in FO4 have Vertibirds (which as far as we know only the D.C. BoS have), the blimps (which the D.C. BoS don't have), and motivation enough to apparently attack the Institute is what says to me that it's a group of BoS which includes the D.C. chapter (but is not limited to them), and that Lyons is either dead or sidelined.
Fallout 3 implied heavily that BoS and the Outcasts were on the brink of war and the only thing that prevented the BoS from escalating the conflict was Lyons and some of the paladins. Oh, and the whole enclave thing. In fact if the BoS and outcast patrols meet eachother, they will engage in combat. So tensions are high and the situation is volatile.
I expect Fallout 4 will expand on this conflict if they choose to include the DC chapter.
So the primary reason the Outcasts split off from the D.C. BoS is simultaneously the reason they're not at war with each other? You do remember why the divide happened, right? Lyons decided to go full charitymodo and part of the chapter didn't like that. Lyons and his worldview is the one thing (apart from the tensions caused by the separation) which split the chapter. Remove him from the picture, give them a couple years under a new leader, and I can easily see the factions reuniting. Even more so if an external group of BoS came through and cleaned house -- if the D.C. chapter didn't go back to their old ways, it's equally possible that an external group of BoS helped the Outcasts retake the Pentagon and the Outcasts are the D.C. chapter now.
What I meant to say was that the only thing preventing BoS from going all-out-war with the Outcasts is Lyons and some of his paladins keeping the BoS under their leash and the Outcasts having some measure of respect for the old coot. Remember, the Brotherhood is very much based on pseudo-knightly nonsense (hurray president Eden) and while the Outcasts dislike Lyons' worldview, they still have a lot of respect for him because he is a very experienced elder who has led the brotherhood through some tough times.
Listening to what the senior brotherhood scribes and paladins have to say about the outcasts paints an image of a relationship that while unfriendly is not entirely hostile. They understand why the Outcasts made their decision and even have a certain measure of respect for them, but in the end, the Outcasts chose wrong and the only reason they seem to restrain themselves from striking against the Outcasts outright is the fact that Lyons forbids it.
Now, the Outcasts aren't nearly as well characterized as the Brotherhood since you are only given very few non-dismissive Outcasts to talk with, but given the way senior members of the Brotherhood seem to feel about the Outcasts it is not too difficult to imagine the Outcasts feeling much the same way and having the same respect for Lyons prevent them from open war (remember, the outcasts are comprised entirely of former members of the brotherhood who were heavily indoctrinated into the dogma).
Once Lyons dies, so does the respect the Outcasts have for him and the conflict escalates. Similarily, the BoS due to its influx of newly recruited members who aren't as well indoctrinated as the old guard, has a much harder time containing the desire for war. They cannot reconcile with the Outcasts yet neither side can tolerate eachother's existence, thus leading into an all-out war.