Minor correction: In your definition of

, you wrote

, and it should have been

.

Also, while it's fairly obvious, you probably should have noted somewhere along the way that any symmetric spinor

is

completely defined by the components

.

These happen to be the same components that appear (individually) in the coefficients of

, which is what allows us to go from equating the

polynomials and

directly to the required

spinor identity.

That said, you could then simplify the last bit of your proof, from (**) down, by noting that

and thus

which, given my preceding comment, establishes immediately that

, since they are both symmetric spinors that yeild the same polynomial when contracted with

.