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Context

I'm studying spinors in detail as part of research project. I'm working through Cartan's Theory of Spinors [1]. In section 53, A spinor is a Euclidean tensor, Cartan asks us to, "Consider a rotation (or reversal)" defined by $$ \begin{bmatrix} x_1^\prime \\ x_2^\prime \\ x_3^\prime \end{bmatrix} = \begin{bmatrix} R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} $$

In Section 53 (pg 42), Cartan states that $$ (R_{11} -iR_{21})^2 + (R_{12} -iR_{22})^2 + (R_{13} -iR_{23})^2 = 0. $$ I had no idea that this was a property of rotation (or reveral) matrices.

I verified that this is true for an arbitrary from axis-angle rotation matrix [2], $R_\text{rot}$, where $$ R_\text{rot} = \begin{bmatrix} \cos \theta +u_x^2 \left(1-\cos \theta\right) & u_x u_y \left(1-\cos \theta\right) - u_z \sin \theta & u_x u_z \left(1-\cos \theta\right) + u_y \sin \theta \\ u_y u_x \left(1-\cos \theta\right) + u_z \sin \theta & \cos \theta + u_y^2\left(1-\cos \theta\right) & u_y u_z \left(1-\cos \theta\right) - u_x \sin \theta \\ u_z u_x \left(1-\cos \theta\right) - u_y \sin \theta & u_z u_y \left(1-\cos \theta\right) + u_x \sin \theta & \cos \theta + u_z^2\left(1-\cos \theta\right) \end{bmatrix}. $$ However, I have not verified this for reflections, or improper rotations.

Question

Can you prove that each and every element of the orthogonal group (that I believe to include proper rotations, improper rotations, and reflections) has the property $$ (R_{11} -iR_{21})^2 + (R_{12} -iR_{22})^2 + (R_{13} -iR_{23})^2 = 0? $$

Bibliography

[1] Cartan, The Theory of Spinors, 1966, p 42.

[2] https://en.wikipedia.org/wiki/Rotation_matrix#Examples_2

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1 Answer 1

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Cartan wrote \begin{align} x_1'&=\alpha x_1+\beta x_2+\gamma x_3\,,\\ x_2'&=\alpha' x_1+\beta' x_2+\gamma' x_3\,,\\ x_3'&=\alpha'' x_1+\beta'' x_2+\gamma'' x_3\,,\\ \end{align} and $$\tag2 (\alpha-i\alpha')^2+(\beta-i\beta')^2+(\gamma-i\gamma')^2=0 $$ which in your notation is $$ (R_{11}-iR_{21})^2+(R_{\color{red}{\boldsymbol{12}}}^2-iR_{22})^2+(R_{13}-iR_{23})^3=0\,. $$ (I highlighted where I fixed a typo of yours).

The equation (2) is obvious because the rows of the matrix are orthonormal so that (2) can be written as \begin{align} \underbrace{\alpha^2+\beta^2+\gamma^2}_{1}-\underbrace{\alpha'\,^2+\beta'\,^2+\gamma'\,^2}_{1}-2i\underbrace{(\alpha\alpha'+\beta\beta'+\gamma\gamma')}_{0}=0\,. \end{align}

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