+ We can change the generator prefix::
+
+ sage: RealSymmetricEJA(3, prefix='q').gens()
+ (q0, q1, q2, q3, q4, q5)
+
+ Our inner product satisfies the Jordan axiom::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = RealSymmetricEJA(n)
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
+ True
+
+ Our basis is normalized with respect to the natural inner product::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = RealSymmetricEJA(n)
+ sage: all( b.norm() == 1 for b in J.gens() )
+ True
+
+ Left-multiplication operators are symmetric because they satisfy
+ the Jordan axiom::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: x = RealSymmetricEJA(n).random_element()
+ sage: x.operator().matrix().is_symmetric()
+ True
+