Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 3 / i

Preserving the symmetric bilinear form infinitesimally means $Z^{T}M+MZ=0$. In blocks of sizes $2,2,1$, write $Z$ as a general block matrix and multiply by the given $M$. The condition says that $MZ$ is skew-symmetric. It forces the lower-right entry to be zero, the middle diagonal block to be the negative transpose of the upper-left block, and the two off-diagonal two-by-two blocks to be skew-symmetric. Thus the general element of the Special orthogonal Lie algebra iswhere $A$ is an arbitrary two-by-two matrix and $p,q$ are arbitrary two-component columns. More explicitly, using ten independent complex parameters,Conversely, substituting this matrix into $Z^{T}M+MZ$ gives zero, so there are no further restrictions. In particular $\dim {\mathfrak{so}}_5=4+1+1+2+2=10$.