Block Matrices and SO(4)

Question

I am currently working with the block matrix $\begin{pmatrix} 0 & G \\ - G^T & 0 \end{pmatrix}$ where $G = a^0 + a^j G^j \in SO(4)$ (for some $a^0, a^j$), where $[G^i, G^j]= 2 \epsilon_{ijk} G^k$ furnish a representation of the $su(2)$ algebra.

I am wondering if it is possible to find a orthogonal matrix such that $\begin{pmatrix} 0 & G \\ - G^T & 0 \end{pmatrix} \rightarrow \begin{pmatrix} 0 & \lambda \\ -\lambda & 0\end{pmatrix}$ upon conjugation, where $\lambda$ is proportional to the identity matrix.

Moreover, in the physical problem that I am working on I have the strong suspicion that actually $\lambda^2 = a0^2 + a^j a_j$, but have not been able to proof that yet – my hope is to recover this result once the correct transformation has been found.

EDIT

I went through my calculations again and found out that I have been missing a minus sign in the lower left block.

Answer 1

For any $G$ in SO(4), you can conjugate by $\begin{pmatrix} G^T & \\ & I \end{pmatrix}$ to get $\begin{pmatrix} & I \\ I & \end{pmatrix}$.

To get minus signs in the lower left, I haven't written down all details (namely orthogonality) but the following will probably work if the statement is true. First, conjugate by $\begin{pmatrix} & A \\ -A & \end{pmatrix}$ for a suitable matrix $A$ to get $\begin{pmatrix} & A^2 \\ A^2 & \end{pmatrix}$, and choose $A$ so that $A^2$ is diagonal with two $+1$'s and two $-1$'s. Then conjugate by a permutation matrix to get the desired form.