To make things faster remember that compliment of an element is such an element which is not related to it and LUB of that goes to I and GLB to O.
So, the complement of e must be an element $e^{'}$ such that LUB(e,$e^{'}$)=I and GLB(e,$e^{'}$)=O, the topmost and bottom-most elements of hasse diagram respectively.
Now, candidates to be looked for the complement of e should be all those elements, to which there is no path in the hasse diagram and those are g,c,d and if you check them these 3 are complements of e.
Now why you only look for non-related elements to be the complement?
Say if I assume b to be a complement of e, from the diagram it is clear that eRb. So by lattice laws
LUB(e,b)=b(b $\not= I$ not okay!!) and GLB(e,b)=e(e=O okay!), so b can never be complement of e.