+
+ TESTS:
+
+ The trace inner product is commutative::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element(); y = J.random_element()
+ sage: x.trace_inner_product(y) == y.trace_inner_product(x)
+ True
+
+ The trace inner product is bilinear::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: a = QQ.random_element();
+ sage: actual = (a*(x+z)).trace_inner_product(y)
+ sage: expected = ( a*x.trace_inner_product(y) +
+ ....: a*z.trace_inner_product(y) )
+ sage: actual == expected
+ True
+ sage: actual = x.trace_inner_product(a*(y+z))
+ sage: expected = ( a*x.trace_inner_product(y) +
+ ....: a*x.trace_inner_product(z) )
+ sage: actual == expected
+ True
+
+ The trace inner product satisfies the compatibility
+ condition in the definition of a Euclidean Jordan algebra::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
+ True
+