It was Felix Klein who applied symmetry groups to equations of mathematical physics. For strange reason he did not do it in case of Maxwell equation, so Henry Poincare' has done it in 1904, but for some strange reason has published his work in a bulletin of some mathematical circle of Palermo (I guess, grandfather of don Corleone was his friend). As for Albert Einstein? please, read his works, for some strange reasons he accepted time as the fourth dimension only in 1914. I am afraid, my reply makes everything more complicated than before.

If you have a k- algebra A, where "algebra" means that A is a k-vector space equpiped with a k-bilinear map m:A\times A \to  A (denote m(a,b)=a.b, A may be Lie, associative, or just nothing), then a derivation makes sense: a k-linear map D:A\to A such that D(a.b)=D(a).b+a.D(b) (and it is always a subalgebra of the Lie algebra of endomorphism of A). If A is also finite dimensional, then you have the usual trace of endomorphisms. In this case, you have (almost) all the ingredients for the definition: call pre-Einstein a derivation D such that

- D is semisimple (as endomorphism)

- tr(DE)=tr(E) for all  E derivation of the algebra E

and if your field contains R, then you have all ingredients: you ask that the eigenvalues of D are all reals.

So, the notion MAKES SENSE for a general k-algebra (A,m). I think the reason that people only consider the Lie algebra case is because they use these derivations for constructing manifolds with special geometrical properties, that they know they come from homogeneous spaces.