### Identities NO. 2 : Algebra, some Euler’s work

11Nov07

First identity comes from Lebesgue:

$(a^2+b^2+c^2+d^2)^2 = (a^2+b^2-c^2-d^2)^2 +(2ac+2bd)^2+(2ad+2bc)^2$

and the second one from Euler:

$(a_1 ^2 + a_2^2 + a_3^2+ a_4^2)(b_1 ^2 + b_2^2 + b_3^2+ b_4^2) = (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 + (a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 + (a_1 b_4 + a_2 b_3 - a_3 b_2+ a_4 b_1)^2$