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2-graded polynomial identities for the Jordan algebra of the symmetric matrices of order two

The Jordan algebra of the symmetric matrices of order two over a field $K$ has two natural gradings by $\mathbb{Z}_2$, the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a $\mathbb{Z}_2$-grading on $B_n$, the Jordan algebra of a symmetric bilinear form in a vector space of dimension $n$ ($n=1$, 2, \dots, $\infty$). We call this grading the \textsl{scalar} one since its even part consists only of the scalars. As a by-product we obtain finite bases of the $\mathbb{Z}_2$-graded identities for $B_n$. In fact the last result describes the weak Jordan polynomial identities for the pair $(B_n, V_n)$.