Homework answers / question archive / Show that the map phi: H->M_2(C) defined by phi(a+bi+cj+dk)=(a+b(sqrt -1),c+d(sqrt-1);-c+d(sqrt-1),a-b(sqrt-1)) is a ring homomorphism

Show that the map phi: H->M_2(C) defined by phi(a+bi+cj+dk)=(a+b(sqrt -1),c+d(sqrt-1);-c+d(sqrt-1),a-b(sqrt-1)) is a ring homomorphism

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Show that the map phi: H->M_2(C) defined by
phi(a+bi+cj+dk)=(a+b(sqrt -1),c+d(sqrt-1);-c+d(sqrt-1),a-b(sqrt-1)) is a ring homomorphism. Calculate its kernel and describe its image.

Ps. Here H is the ring of integral Hamilton Quaternions and M_2(C) are 2x2 matrices with complex coefficients.
notation after the equal sign in the definition of phi is a matrix with the first row of two elements a+b(sqrt-1) and c+d(sqrt-1) and second row of two elements
-c+d(sqrt-1) and a-b(sqrt-1)