Of course, the rest of the board ultimately defines the answer.
However, if you have the chance to separate alive groups, and no other exceptionally big moves on the board, do y’all usually go for it or ignore?
I realize this is a very general question. 
I’ve been wondering about that too.
Consider that separating them probably connects two of your groups, so you can double whatever it’s worth.
Also, even if the groups are all “alive”, they won’t usually have solid eyes, and by connecting you reduce the urgency of defending them.
Yeah, it totally makes sense to separate living groups if possible ^^
When separating the groups, you often gain lot of end-game against those groups, and your opponent suddenly has two borders they need to seal by playing dame moves instead just having one continous/connected territory. Those dame moves are moves they could have played elsewhere and potentially gained some points.
And you usually gain a lot of extra ko threats when your opponent has disconnected groups, even if the groups are alive.
But of course it depends, if the groups are already sealed and there isnt any end-game against them, and your cutting is simply gote dame, then yeah maybe its not a good idea xD
so… in most cases its worth at least 2 points of territory, usually more.
I don’t think it helps in the same way as splitting weak groups, but it can be a pretty significant reduction if you get the chance. An artificial example:
White only needs 5 stones to split the groups, but black needs 8 stones to close the boundary.
Separating two groups which are already alive is yose, you need to count to determine if it’s the right timing. In the position below, A and B are equivalent.
Nice and simple example. I think for many player up to SDK ranks the intuition is that B is larger. So I guess the message might be: try to count the points gained and make the largest move. If you can’t and have to rely on your intuition, consider separating, because your intuition is likely to underestimate the value of that.
Isnt A actually better due ko threats? Both are worth 4 points, but if white plays at B and black plays A, white has only one ko threat at J1/L1, because if w does J1 b can just connect on L1 instead taking the stone. But if w chooses A and b plays at B, both J1 and L1 work as ko threats for white, and w even has follow-up threats with H2/G1 and M2/N1?
Tho of course without ko on the board its a moot point xD
If A is the very last point on the board, it’s just 3 points instead of 4 points, since black can take gote and prevent white from getting both J1 and L1.
If the whole-board position is as shown, and it’s white to play, white can choose freely between A and B. But if it’s black to play, the point B is one point better than point A. (assuming territory scoring - in area scoring it would be two points better)
In “normal” circumstances A and B have the same value, because it’s rare that black would want to play a 1-point gote immediately after white plays the big move A (there would usually be other more valuable things to do). So earlier in the game it’s likely that both sides would prefer A, for the ko threat reasons @_KoBa mentioned.
Go is annoying/amazing in that even in such simple examples we cannot give a general answer to which of A and B is better to play first 
