The preposition is with de, embaixo de, debaixo de, abaixo de. The adverbs don't have a de. The adverb doesn't describe the relationship between two objects, it describes the circumstances or the way the action described by the verb of the sentence is preformed.
All thre adverbs contain baixo, wherin baixo is an adjective and means soft, quiet, small, sad. One could think that semantical differences can be deduced from the different prepositions, em, de, a. However this is not the case. I most cases the three are interchangeable.
In the following sentences there is no object and therefore it is easy to understand that down, below, unter, downwards, beneath are not used as prepositions, because prepositions describe the relationship between two objects, but if there is none, they can't be prepositions. They are adverbs in the following sentences
I go down.
I heard someone calling below.
The boat was going under.
They went downwards.
The town is below the castle.
To see what is beneath, you have to look.
All these adverbs describe an action and embrace the same semantic field, but the meaning is not the same. In the sentence "I go down" the focus is more on the final position of the mouvement, for instance the cellar. If we go downwards, the focus is on the direction. If we cycle downwards for instance we focus on the fact that less energy is to be spent and where we end up is not so important. In the sentence "To see what is beneath...", beneath is actually a pronominal adverb. Beneath refers to something, the bed, the table, the cover. In the case of the city, that is below the castle, it is not vertically below the castle, but horizontally. If someone is calling below, he doesn't move. To summarise that: All these adverbs belong to the same semantic field, but they are not interchangeable and describe different aspects. Some describe a movement to a certain place, others describe a direction in the three dimensional space, some are prenominal adverbs and refers to something, some decribe a vertical relationships and others a horizontal relationships. The same distinctions to be made in English has to be made in Portuguese as well. If the reader wants to understand how difficult it is to distill some rule concerning the use of the Portuguese adverbs, he can try to figure out the rule concerning the use of the English adverbs. In other words: It is somehow confusing in any language, this is not a portuguese speciality.
It is hard to find rules or explanations concerning the use of abaixo, embaixo, debaixo from "oficial" linguists, in other words from people who feel for whatever reason a calling to discuss linguistic issues and whose judgement can be considered as qualified. This author would say that the following is more or less true, although the actual use is waffering and there is a large debate even between Portuguese native speakers.
As an adverb abaixo means, downwards, downstairs, below, down, beneath. As an adverb it can only refer to a verb, can only describe the circumstances the action decribed by the verb is performed. Furthermore only abaixo can be used in a methaphorical meaning, can for instance describe a mental state. Abaixo can be combined with other prepostions.
I would not allow myself to be beaten down so easily.
Então resolveram usar uma broca para ver o que havia abaixo.
They decided to use a drill in order to see what is below.
Espera até ele chegar aqui abaixo.
Wait until he comes down.
Podem acompanhá-lo até abaixo.
You can accompany him downstairs.
Há um erro duas linhas abaixo.
Two lines below there is an error.
Embaixo can be an adverb, but is mostly used together with another preposition. Exceptions are the use after copulative verbs like estar (to be / to be located), ficar (to be / to remain).
(Copulative verb: Verb that connects a property to a noun: He is ill. He becomes rich. He gets angry. And so on.)
embaixo, até embaixo* (downstairs), aqui embaixo (down here), lá embaixo (down there)
Vou procurar lá embaixo.
I will look for it downstairs.
Venha aqui embaixo.
Come down here.
Olhando do décimo andar de um prédio tudo parece pequeno lá embaixo.
Looking down from the tenth floor of a building, everything seems small down there.
A oficina fica embaixo.
The office is downstairs.
* In Brasil it is written in one word, embaixo, in Portugual it is separated, em baixo. In Brasil it is only written separatedly if baixo is an adjective: falar em baixo tom de voz <=> to speak in a low voice.
As prepositions embaixo de and debaixo de have the same semantic value.
debaixo beneath*, under, below
Moro no terceiro andar e ela mora no apartamento debaixo.
I live in the third floor and she live in the floor beneath.
Levantou o pano para ver o que havia debaixo.
He lifted the cloth to see what is beneath.
Debaixo estava um homem.
Below there was a man.
* If we want to be precise, beneath is an pronominal adverb. If we don't know the context in a sentence like "He looked beneath", everybody would ask beneath what.