You can try comparing the ingredient list to other products that you regularly use, like body lotion, hair products, etc. This will allow you to rule out the more common ingredients that you know you can tolerate. Then you can have a lot smaller list of potential allergens to choose from.
Try to find the information on internet as well to gain knowledge
You can search for Allergic reaction for example,
http://www.fealth.com/health/search?q=Allergic+reaction
Also, try to check for the creator or reviewer in each of these web pages.