Aloe Vera Gel- your aloe
Triethanolamine- emulsifier, it is somewhat basic in nature and this high of a concentration is likely the source of your irritation
Tocopheryl Acetate- vitamin e
Carbomer- thickener
Tetrasodium EDTA-preservative and chelating agent
DMDM Hydantoin-preservative, the irritating kind
Diazolidinyl Urea-preservative, it's the more irritating kind (parabens would have been a better choice)

Because the types of preservatives in this product are the type that are less preferable to parabens in terms of irritation potential, they are possibly the cause of your irritation. But more likely it's the triethanolamine, which is used to make sure all the different ingredients in the product stay together and don't separate. It is somewhat basic in terms of it's PH, which can be harsh on delicate skin depending on the concentration. It's in almost every product on the market in low below 1% concentrations without causing irritation for the majority of people, but because it's second on the list that tells me it's more than likely more than 1%. The fact that it's irritating your eyes and nothing else tells me that it's just sensitivity and not allergy, so I don't suggest avoiding it entirely. Just look out for products that have a lot and don't use those specific ones around the eyes.