I do not believe that when one member of an alliance declares war on another, it should be incumbent upon the other alliance members to participate as belligerents as well.
That's how it works in real life. If you don't want to go to war, you can always leave the alliance.
It depends upon the type of alliance and the terms of the Treaty ratifying the alliance, both requiring further clarification.
However, real life history is replete with examples of alliances where while one member may have decided to go to war, without the others being obligated to follow suit.
1) NATO (and Warsaw Pact) - The Parties of NATO are only obligated to assist a particular member should that member be attacked, and even should that occur, it is definitely not incumbent upon them to launch their own direct counter-offensive against the aggressor.
2) The Triple Entente (World War I) - This military alliance between Britain, France and Russia did not oblige any of its members to participate in a war declared by another; in fact, the Great Powers all had their own underlying motives to initiate hostilities against their European enemies and were not necessarily obliged to become belligerents: Russia wished to protect Serbia and the Slavs, France used this as an opportunity to re-take Alsace and Lorraine whereas Britain's cause to go to war was because Germany invaded Belgium and thus violated Belgian neutrality.
3) The Triple Alliance (World War I) - When Germany and Austria-Hungary declared war against the Allies, Italy, while remaining a member of the alliance and pledging support to her partners, remained neutral in the war, stating that it was not obligated to participate in a conflict initiated by Germany and Austria-Hungary.
4) The Auld Alliance - Signed between Scotland and France, this Treaty stipulated that should one country ever be attacked by England, that the other country would invade the agressing nation. Once again, the obligation to go to war with one's allies is not present; if Scotland invaded England, it was not incumbent upon France to do the same.
5) MNNA (Major Non-Nato Ally) - It is a designation used by the United States Government to refer to countries whom, while not being members of NATO, possess very strong and significant ties with the US. Note, however, that these MNNAs were by no means obligated to support any American offensive, in Afghanistan or in Iraq, and note also that this, by no means, entails the stripping of their special status and the ending of the special relation.
I am certain that more examples exist, but I have listed above what my thoughts immediately turned to.