We know God’s essence and existence are the same thing. God necessarily exists; existence necessarily exists. We know God is not made of stuff, has no potentiality, is not made of parts (is “simple” in a technical sense). He has no extraneous properties. Let’s continue to flesh out, as it were, more of God. We’ll skip a little quickly through these two chapters, because they’re easier and non-controversial (everything here follows simply if you accept what came before). The reader is encouraged (do it!) to go to the original chapters for the complete story.
Chapter 24: That the divine being cannot be specified by the addition of any substantial difference
2 …Whatever needs something added to it, in order to exist, is in potentiality to that thing. But the divine substance is not in potentiality in any way, as proved above: and God’s substance is His being. Therefore His being cannot receive essential specification from something added to it.i
3 Moreover. Whatever makes a thing to be in act, and is intrinsic to that thing, is either the whole essence thereof or part of its essence. Now that which specifies a thing by an essential specification, makes a thing to be in act, and is intrinsic to the thing specified: otherwise the latter could not be specified essentially thereby.ii Therefore it must be either the very essence or part of the essence of that thing. But if something be added to the divine being, it cannot be the whole essence of God, for it has already been proved that God’s existence is not distinct from His essence. Therefore it follows that it is a part of the divine essence: and thus God would be composed of essential parts, the contrary of which was proved above.…
Chapter 25: That God is not in any genus
1 HENCE it follows of necessity that God is not in any genus.iii For whatever is in a genus, has in itself something whereby its generic nature is specified: for nothing is in a genus without being in some one of its species. But in God this is impossible, as shown above.[1′] Therefore it is impossible that God be in any genus…
3 Again. Whatever is in a genus differs as to being from the other things contained in the same genus: otherwise a genus would not be predicated of several things. Now all things that are contained in one same genus, must agree in the whatness of the genus, because the genus is predicated of all in respect of what a thing is. Therefore the being of anything contained in a genus is beside the whatness of the genus. But this is impossible in God.[4′] Therefore God is not in a genus.
4 Further. A thing is placed in a genus by the nature of its whatnessiv, for genus is predicated of what a thing is. But the whatness of God is His very being.[5′] Now a thing is not placed in a genus according to its being, because then being would be a genus signifying being itself.v It remains therefore that God is not in a genus.
5 That being cannot be a genus is proved by the Philosopher as follows.[6′] If being were a genus, it would be necessary to find a difference in order to contract it to a species. Now no difference participates in the genus, so that, to wit, the genus be contained in the notion of the difference, for thus the genus would be placed twice in the definition of the species: but the difference must be something besides that which is contained in the notion of the genus.vi Now there can be nothing besides that which is understood by being, if being belong to the notion of those things of which it is predicated. And thus by no difference can being be contracted. It remains, therefore, that being is not a genus: wherefore it follows of necessity that God is not in a genus.
iIf you need food, then you’re potentially fed. (Something actual must bring this about.) But if you need something that you don’t have, you’re not complete. And God is complete, and has no potentiality (and no parts).
iiThis two sentences are, after a moment’s thought, obvious. But look how well they’re put. Succinct city! It’s also good to reflect. That which causing something to be actual is what makes up that thing’s essence. Causality (act) and essence are linked. This applies everywhere, of course. Not just to God.
iiiFor Aristotle and Aquinas, there are two levels, species and genus, the latter being a collection (if you like) of species. Genus is higher up in the taxonomic order. If something is in a genus, it must then be in one of the species of that genus. But God is sui generis. There is nothing like Him.
vEvery being has being; rather, everything that is in existence, so that it exists does not a genera make.
viThe key phrase is “the difference must be something besides that which is contained in the notion of the genus.” You cannot use the definition of genus as a genus. It is to be real differences which make up a genus. The rest is like the previous argument.
 Ch. xvi.
 Ch. xxii.
 Ch. xviii.
[1′] Ch. xxiv.
[4′] Ch. xxiv.
[5′] Ch. xxii.
[6′] Metaph. iii. 8.