+ def cartesian_inner_product(self, x, y):
+ r"""
+ The standard componentwise Cartesian inner-product.
+
+ We project ``x`` and ``y`` onto our factors, and add up the
+ inner-products from the subalgebras.
+
+ SETUP::
+
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLE::
+
+ sage: J1 = HadamardEJA(3,field=QQ)
+ sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: x1 = J1.one()
+ sage: x2 = x1
+ sage: y1 = J2.one()
+ sage: y2 = y1
+ sage: x1.inner_product(x2)
+ 3
+ sage: y1.inner_product(y2)
+ 2
+ sage: z1 = J._cartesian_product_of_elements((x1,y1))
+ sage: z2 = J._cartesian_product_of_elements((x2,y2))
+ sage: J.cartesian_inner_product(z1,z2)
+ 5
+
+ """
+ m = len(self.cartesian_factors())
+ projections = ( self.cartesian_projection(i) for i in range(m) )
+ return sum( P(x).inner_product(P(y)) for P in projections )
+
+