+ return Qs
+
+
+def _embed_complex_matrix(M):
+ """
+ Embed the n-by-n complex matrix ``M`` into the space of real
+ matrices of size 2n-by-2n via the map the sends each entry `z = a +
+ bi` to the block matrix ``[[a,b],[-b,a]]``.
+
+ EXAMPLES::
+
+ sage: F = QuadraticField(-1,'i')
+ sage: x1 = F(4 - 2*i)
+ sage: x2 = F(1 + 2*i)
+ sage: x3 = F(-i)
+ sage: x4 = F(6)
+ sage: M = matrix(F,2,[x1,x2,x3,x4])
+ sage: _embed_complex_matrix(M)
+ [ 4 2| 1 -2]
+ [-2 4| 2 1]
+ [-----+-----]
+ [ 0 1| 6 0]
+ [-1 0| 0 6]
+
+ """
+ n = M.nrows()
+ if M.ncols() != n:
+ raise ArgumentError("the matrix 'M' must be square")
+ field = M.base_ring()
+ blocks = []
+ for z in M.list():
+ a = z.real()
+ b = z.imag()
+ blocks.append(matrix(field, 2, [[a,-b],[b,a]]))
+ return block_matrix(field, n, blocks)
+
+
+def RealSymmetricSimpleEJA(n):
+ """
+ The rank-n simple EJA consisting of real symmetric n-by-n
+ matrices, the usual symmetric Jordan product, and the trace inner
+ product. It has dimension `(n^2 + n)/2` over the reals.
+ """
+ pass
+
+def ComplexHermitianSimpleEJA(n):
+ """
+ The rank-n simple EJA consisting of complex Hermitian n-by-n
+ matrices over the real numbers, the usual symmetric Jordan product,
+ and the real-part-of-trace inner product. It has dimension `n^2 over
+ the reals.
+ """
+ pass
+
+def QuaternionHermitianSimpleEJA(n):
+ """
+ The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
+ matrices, the usual symmetric Jordan product, and the
+ real-part-of-trace inner product. It has dimension `2n^2 - n` over
+ the reals.
+ """
+ pass
+
+def OctonionHermitianSimpleEJA(n):
+ """
+ This shit be crazy. It has dimension 27 over the reals.
+ """
+ n = 3
+ pass
+
+def JordanSpinSimpleEJA(n):
+ """
+ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
+ with the usual inner product and jordan product ``x*y =
+ (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
+ the reals.
+ """
+ pass