(This is an answer to the updated question). I have a bit trouble with your terminology (what you are talking about are definitely not direct sums of a Lie and a Jordan algebra), but I hope I understand the spirit of your question, without going to much into nitpicking. Again, being put, probably, in a bit more general framework, the question is whether the algebras with a few binary operations, one of them, for example, satisfies the Jacobi identity, another one satisfies the Jordan identity, with some compatibility conditions between them, or something like that, were considered in the literature.
The answer is "yes". In fact, there is so much papers about such sort of objects, that it is a bit difficult to point just on a few of them. Lot of such objects are studied nowadays in the framework of operadic theory, see a very interesting compendium arXiv:1101.0267 . Some authors who wrote a lot on the topic: Dzhumadil'daev (e.g., https://doi.org/10.1142/S0219498809003230 , https://doi.org/10.1007/s10958-009-9532-x , http://www.mpim-bonn.mpg.de/preblob/2920 , http://dx.doi.org/10.1081/AGB-200060504 ), M. Goze, Markl and Remm (look them on arXiv).
