Naive Lie Theory by John Stillwell 1.3.6 Show that the multiplicative property of determinants gives the real four-square identity(af + b} +c}+d})(az + b3 + c3 + d}) = (aja2 – bīb2 – c1C2 – dịd2)?+ (a¡b2+bja2+cjd2 – djc2)²+ (ajc2 – bịd2 +cja2+djb2)²+ (ajd2+ bịc2 – cıb2+dja2)².This identity was discovered by Euler in 1748, nearly 100 years before the dis-covery of quaternions! Like Diophantus, he was interested in the case of integersquares, in which case the identity says that(a sum of four squares) × (a sum of four squares) = (a sum of four squares).This was the first step toward proving the theorem that every positive integer isthe sum of four integer squares. The proof was completed by Lagrange in 1770.

Answered: 1.3.6 Show that the multiplicative…

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Naive Lie Theory by John Stillwell

1.3.6 Show that the multiplicative property of determinants gives the real four-
square identity
(af + b} +c}+d})(az + b3 + c3 + d}) = (aja2 – bīb2 – c1C2 – dịd2)?
+ (a¡b2+bja2+cjd2 – djc2)²
+ (ajc2 – bịd2 +cja2+djb2)²
+ (ajd2+ bịc2 – cıb2+dja2)².
This identity was discovered by Euler in 1748, nearly 100 years before the dis-
covery of quaternions! Like Diophantus, he was interested in the case of integer
squares, in which case the identity says that
(a sum of four squares) × (a sum of four squares) = (a sum of four squares).
This was the first step toward proving the theorem that every positive integer is
the sum of four integer squares. The proof was completed by Lagrange in 1770.

Expert Solution

Step 1

To show:

The multiplicative property of determinants gives the real four square identity: -

$\begin{array}{rcl} ({a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2} + {d_{1}}^{2}) ({a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2} + {d_{2}}^{2}) & = & {(a_{1} a_{2} - b_{1} b_{2} - c_{1} c_{2} - d_{1} d_{2})}^{2} + {(a_{1} b_{2} + b_{1} a_{2} + c_{1} d_{2} - d_{1} c_{2})}^{2} \\ + {(a_{1} c_{2} - b_{1} d_{2} + c_{1} a_{2} + d_{1} b_{2})}^{2} + {(a_{1} d_{2} + b_{1} c_{2} - c_{1} b_{2} + d_{1} a_{2})}^{2} \end{array}$