+ Rank is additive on a Cartesian product::
+
+ sage: J1 = HadamardEJA(1)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J1.rank.clear_cache()
+ sage: J2.rank.clear_cache()
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 3
+ sage: J.rank() == J1.rank() + J2.rank()
+ True
+
+ The same rank computation works over the rationals, with whatever
+ basis you like::
+
+ sage: J1 = HadamardEJA(1, field=QQ, orthonormalize=False)
+ sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: J1.rank.clear_cache()
+ sage: J2.rank.clear_cache()
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 3
+ sage: J.rank() == J1.rank() + J2.rank()
+ True
+
+ The product algebra will be associative if and only if all of its
+ components are associative::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J1.is_associative()
+ True
+ sage: J2 = HadamardEJA(3)
+ sage: J2.is_associative()
+ True
+ sage: J3 = RealSymmetricEJA(3)
+ sage: J3.is_associative()
+ False
+ sage: CP1 = cartesian_product([J1,J2])
+ sage: CP1.is_associative()
+ True
+ sage: CP2 = cartesian_product([J1,J3])
+ sage: CP2.is_associative()
+ False
+
+ Cartesian products of Cartesian products work::
+
+ sage: J1 = JordanSpinEJA(1)
+ sage: J2 = JordanSpinEJA(1)
+ sage: J3 = JordanSpinEJA(1)
+ sage: J = cartesian_product([J1,cartesian_product([J2,J3])])
+ sage: J.multiplication_table()
+ +--------------++---------+--------------+--------------+
+ | * || e(0, 0) | e(1, (0, 0)) | e(1, (1, 0)) |
+ +==============++=========+==============+==============+
+ | e(0, 0) || e(0, 0) | 0 | 0 |
+ +--------------++---------+--------------+--------------+
+ | e(1, (0, 0)) || 0 | e(1, (0, 0)) | 0 |
+ +--------------++---------+--------------+--------------+
+ | e(1, (1, 0)) || 0 | 0 | e(1, (1, 0)) |
+ +--------------++---------+--------------+--------------+
+ sage: HadamardEJA(3).multiplication_table()
+ +----++----+----+----+
+ | * || e0 | e1 | e2 |
+ +====++====+====+====+
+ | e0 || e0 | 0 | 0 |
+ +----++----+----+----+
+ | e1 || 0 | e1 | 0 |
+ +----++----+----+----+
+ | e2 || 0 | 0 | e2 |
+ +----++----+----+----+
+