I'm not sure what kind of information is being asked for in the question, but Claudio has offered a coherent viewpoint based on some explicit Lie algebra theory. It's also possible to work on the level of internal structure, given by Dynkin (and extended Dynkin) diagrams, etc. Chapter X of the standard book by Helgason Differential Geometry, Lie Groups and Symmetric Spaces (partly influenced by work of Kac) draws together a lot of the classification-related material you might need.
Embeddings of the Lie algebra of type $B_4$ into the Lie algebra of type $F_4$ occur naturally in terms of the extended Dynkin diagram for $F_4$ (page 503 of Helgason, for example). Here you see the diagram of $B_4$: for a fixed Cartan subalgebra of the bigger algebra, you get simple roots for the subalgebra sharing this Cartan subalgebra by taking the two long simple roots and adjacent short simple root together with the negative of the highest (long) root of $F_4$ which occurs as the extra node in the extended diagram. What you have here is not a Levi subalgebra of a standard parabolic in $F_4$ but rather a pseudo-Levi subalgebra.
From the internal viewpoint of the adjoint representation, this construction is entirely explicit. Moreover, beyond the arbitrary choice of a common Cartan subalgebra, the finitely many automorphisms of the bigger or the smaller Lie algebra stabilizing this Cartan subalgebra seem to provide the only variation in the embedding. Naturally you might prefer to realize all of this in terms of Jordan algebras and the like, but it's already visible in the abstract Lie algebras.
For the comparison of Lie algebras of types $F_4$ and $E_6$, probably the most natural Lie algebraic direction is the "folding" suggested by Johannes. This too is treated in Helgason's chapter and in the book by Kac, as well as in algebraic group classifications. Here you realize the smaller Lie algebra as a sort of quotient of the bigger one by identifying certain simple roots of $E_6$ according to the natural Dynkin diagram automorphism. But if you want to go the other way and get a natural embedding, it's probably best to follow Claudio's construction. Again I don't see more than one way to do this abstractly, but you can always apply some compatible automorphisms of the Lie algebras.