Bourbaki

Exercise 19, § 2

Algebra, III, 624

Prove the polynomial identity in 8 indeterminates

(X12+X22+X32+X42)(Y12+Y22+Y32+Y42)=(X1Y1−X2Y2−X3Y3−X4Y4)2+(X1Y2+X2Y1+X3Y4−X4Y3)2+(X1Y3+X3Y1+X4Y2−X2Y4)2+(X1Y4+X4Y1+X2Y3−X3Y2)2. (X_1^2 + X_2^2 + X_3^2 + X_4^2)(Y_1^2 + Y_2^2 + Y_3^2 + Y_4^2) = (X_1 Y_1 - X_2 Y_2 - X_3 Y_3 - X_4 Y_4)^2 + (X_1 Y_2 + X_2 Y_1 + X_3 Y_4 - X_4 Y_3)^2 + (X_1 Y_3 + X_3 Y_1 + X_4 Y_2 - X_2 Y_4)^2 + (X_1 Y_4 + X_4 Y_1 + X_2 Y_3 - X_3 Y_2)^2.

(Apply formula (32) of no. 5 to a suitable quaternion algebra.)

Solution

No solution has been written yet.

Transcribed from the printed volume. Not translated.