SR and LT

The Lorentz group consists of linear transformations Λ that leave $x^2=x_{\mu }{x}^{\mu }$ invariant. They have the form $x^{\mu }⟶x^{\prime \mu }={\Lambda }_{\nu }^{\mu }{x}^{\nu }{x}_{\mu }⟶x_{\mu }^{\prime }={\Lambda }_{\mu }^{\nu }{x}_{\nu }=x_{\nu }({\Lambda }^{-1}{)}_{\mu }^{\nu }$ For LTs, the norm is invariant so that, $x^2\to (x^{\prime }{)}^2=(x^{\prime }{)}^{T}G{x}^{\prime }=x^T{\Lambda }^{T}G\Lambda x=x^{T}Gx\text{ iff }{\Lambda }^{T}G\Lambda =G$ The Lorentz group splits into 4 components: \text{i) }\Lambda = \mathbb{1}$$$$\text{ii) }\Lambda=-G$$$$\text{i) }\Lambda = -\mathbb{1}$$$$\text{ii) }\Lambda=G These four elements form a subgroup of dimension 4.