Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring
or P(B|A) = P(B)
Coming to our example,
As you had mentioned earlier,
-> the sum of the dice is 6
F -> the sum of the dice is 7
G -> the first die rolled is a 4
F is a special case because we have 6 different ways of getting a sum of 7 unlike any other number.(Think of the sample space for any other sum.)
Lets look at the sample space of F
F = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
Notice that, irrespective of what the number is on the first or second die, the probability of getting a sum of seven is a constant.
This isn't the case for event E (the event that the sum of the dice is 6), let's confirm the sample space
E = {(1,5), (2,4), (3,3), (4,2), (5,1)}
Notice that, the event of getting a 6 on either die excludes the possibility of getting a sum of 6 and hence our probability is influenced by/dependent on that.
This, IMHO, is the intuitive reason for the independence of events F and G.