Again, these are contrived, in my opinion. Just give me a naturally occuring lie algebra that is not gotten from simply considering an associative algebra or a subspace of some larger infinite dimensional in some cases associative algebra if possible. I can't think of one; this isn't a test; it's a genuine query; there is nothing that requires the product of two elements in a lie algebra to exist, merely their bracket, though trivially we can declare them to exist, just as we can formally declare lots of things to exist. For those with such a mind, what about vector fields and the lie derivative?
