# Definition The restricted Lorentz group in $n$ dimensions $SO(n-1, 1)_o$ is the [[Identity Component]] of the [[Lorentz Group]]. # Lie Algebra The [[Lie Algebra]] of $O(n-1,1)$, $\mathfrak{o}(n-1,1)$, is the set shown [[Lorentz Group#Lie Algebra mathfrak o n-1 1|here]]. Imposing the unit determinant condition we get: $ \det{e^{tX}} = e^{t \text{tr}(X)} = 1 \implies \text{tr}(X) = 0 $ However, these matrices are already traceless, thus, we conclude that $\mathfrak{so}(n-1,1) = \mathfrak{o}(n-1,1)$. We will always refer to them as $\mathfrak{so}(n-1,1)$. This is not difficult to understand. The Lie algebra is the [[tangent space]] of the [[Lie group]] at the identity element. In the case of $O (n-1,1)$, the identity element is contained in the [[Identity Component]], i.e., $SO(n-1,1)$. This is why we have $\mathfrak{o}(n-1,1) = \mathfrak{so}(n-1,1)$ Note that $\mathfrak{o}(n-1,1) = \mathfrak{so}(n-1,1)$, the Lie algebra of the [[Restricted Lorentz Group]]. See [[Restricted Lorentz Group#Lie Algebra|here]] for a discussion.