 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2014/4/e/ii/solution

The tangent space at the identity is the kernel of a linear map from part (i):

T_{I} G = {H : H^{T} J + J H = 0} .

This says precisely that

J H

J^{- 1} = J

J

identifies this tangent space with the three-dimensional space of skew-symmetric

3 \times 3

matrices. In entries it is

H = a c 0 b 0 - c 0 - b - a, a, b, c \in R .

Substitution verifies the equation directly and displays its three independent parameters.

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