+ """
+ Faster to reimplement than to use natural representations.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
+
+ TESTS:
+
+ Ensure that this is the usual inner product for the algebras
+ over `R^n`::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = RealCartesianProductEJA(n)
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: X = x.natural_representation()
+ sage: Y = y.natural_representation()
+ sage: x.inner_product(y) == J.__class__.natural_inner_product(X,Y)
+ True
+
+ """
+ return x.to_vector().inner_product(y.to_vector())