REBT Perspective on the problem: This is an all too common problem that sadly people seek help in solving. The first thing to do is to help yourself before you attempt to help anyone else involved whether that be your Mother, brother, or sister.
REBT teaches that we upset ourselves about what happens to us but also teaches you can choose between healthy and unhealthy negative emotions. If you are experiencing anxiety, anger, shame, or depression in response to what is going on here you are experiencing an unhealthy negative emotion. If you are experiencing concern, annoyance, disappointment or sadness you are experiencing healthy negative emotions. The unhealthy negative emotions will interfere with any effort you make to repair this rift in the family while the healthy negative emotions will help motivate you to creatively intervene to try to influence your fallible siblings. Furthermore, whether you are successful in repairing the relationship between your siblings your unhealthy negative emotion will make harder for you to experience well-being and some degree of happiness while your siblings are at war with each other. So from an REBT standpoint unhealthy negative emotion is self-defeating for many different reasons. Give them up! But how you ask? The answer lies with you striving to hold flexible and non-extreme attitudes about the rift that is occurring. These healthy attitudes would be “I wish my brother and sister would not anger themselves with each other but they do not have to be sane, tolerant, and flexible as I want them to be. They are both fallible humans and right now are acting as they are acting because that is their nature. Now I had better accept them warts and all and model tolerance for both of them and this is probably the best thing I can do to help this sad situation. I will not disturb myself about their anger towards each other and I will not disturb myself if my mother is sad about their anger towards each other. She has a right to be sad as I do because we want them to get along or at least tolerate each other and we are NOT getting that right now and may very well never get that. Too bad!”
This rational philosophy will help you feel sad and concerned two emotions which will serve to motivate you to attempt to influence your brother and your sister. When you take steps to speak to them about their anger for the other party see that the only person you can control is you. REBT teaches that you cannot control anyone else other than you. If you are lucky you may find a way to influence them somewhat but in the end humans will do what they want and not what you want. Accept this.
When you speak to them you may want to sneak in some teachings from REBT. First when you discuss their upset with the other party you may not want to try to persuade them to not see the other person as having not committed a transgression. Instead accept that their reality is “true” and ask them if they might not be better off condemning the sin committed rather than condemning the sinner. Ask them if their feelings towards the other are serving them well. Is their anger and resentment helping them or hurting them in any way? They may not be inclined to see their feelings as hurting them but anger is very self-defeating and you may want to gently point out to them how their anger is hurting them. Sadly, you may not get very far no matter how logical you argument is and how badly their anger is hurting them. Model unconditional other-acceptance and do not demand that they relinquish their anger. You can ask but people do not have to give you what you want and so you may very well have to live with your brother and sister fighting and sulking like cats and dogs. This is very sad but not the bloody end of the world. Keep control of your attitudes and reactions towards each of them and model to the day you die the unconditional-other acceptance REBT teaches.
As for your elderly mother strive to not disturb yourself over her reaction to the rift between your brother and sister. You can again sneak in some of the REBT philosophy by showing her that sadness, sorrow, disappointment, and concern are sane and healthy for her to feel while depression, shame, guilt, and anxiety are very unhealthy for her to feel. Encourage her to unconditionally accept both of her children while not being at all pleased with their behavior towards one another. If she feels guilty in some way encourage her to go forward unconditionally accepting herself but to put down any contribution she may have had to this problem. For example, did she do something similar in her relationships over the years? If so that is bad but she too is not a bad person. Finally, encourage her to have unconditional life-acceptance and see that these sorts of things have happened for thousands of years between siblings and are likely to continue to occur as long as fallible humans inhabit the planet. She can still have some happiness despite her disappointment that her children are needlessly disturbing themselves with each other. We all can try to have a positive influence on the situation but not disturb ourselves when we do not make as much headway as we would like to make.
Back to you and your problem. Remember to strive to have some happiness with each sibling even though you probably would derive more happiness if your siblings were part of one big happy family. Too bad we are all fallible humans who sometimes get stuck on anger.