Suppose I'm given an operation on a set such that the operation between two elements of set is also an element of set

Now there are possible binary operations of on set .

If I'm asked to find if there exists a neutral/identity element of a certain table where

are defined, how do I figure that out?

What I mean is suppose 1 of 16 table of operations looks like this:

How do I figure out if an identity element exist for this table? Also is it possible to find the identity/neutral element from this table?