Lorentz transformation

Given that the interval ${\displaystyle x^{\mu }x_{\mu }}$ is invariant under a Lorentz transformation, prove that the Lorentz transformation is orthogonal.

1.	${\displaystyle x'^{\mu }x'_{\mu }=x^{\alpha }x_{\alpha }}$.	Given.
2.	${\displaystyle x'^{\mu }x'_{\mu }=g_{\mu \nu }x'^{\mu }x'^{\nu }}$	metric tensor
3.	${\displaystyle x'^{\mu }=\Lambda ^{\mu }{}_{\alpha }x^{\alpha }}$	Lorentz transformation
4.	${\displaystyle x'^{\nu }=\Lambda ^{\nu }{}_{\beta }x^{\beta }}$	Lorentz transformation
5.	${\displaystyle x'^{\mu }x'_{\mu }=g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }x^{\alpha }\Lambda ^{\nu }{}_{\beta }x^{\beta }}$	Substitute 3 and 4 into 2.
6.	${\displaystyle x^{\alpha }x_{\alpha }=g_{\alpha \beta }x^{\alpha }x^{\beta }}$	metric tensor
7.	${\displaystyle g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }x^{\alpha }\Lambda ^{\nu }{}_{\beta }x^{\beta }=g_{\alpha \beta }x^{\alpha }x^{\beta }}$	From 5, 1, and 6.
8.	${\displaystyle g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }\Lambda ^{\nu }{}_{\beta }x^{\alpha }x^{\beta }=g_{\alpha \beta }x^{\alpha }x^{\beta }}$	Rearrange 7.
9.	∴ ${\displaystyle g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }\Lambda ^{\nu }{}_{\beta }=g_{\alpha \beta }}$	From 8, since ${\displaystyle x^{\alpha }}$ and ${\displaystyle x^{\beta }}$ may be arbitrary.
10.	${\displaystyle g_{\alpha \beta }g^{\alpha \gamma }=\delta ^{\gamma }{}_{\beta }}$	Kronecker delta
11.	${\displaystyle g_{\mu \nu }\Lambda ^{\mu }{}_{\alpha }\Lambda ^{\nu }{}_{\beta }g^{\alpha \gamma }=\delta ^{\gamma }{}_{\beta }}$	Multiply both sides of 9 by ${\displaystyle g^{\alpha \gamma }}$, then apply 10.
12.	${\displaystyle g_{\mu \nu }\Lambda ^{\mu \gamma }\Lambda ^{\nu }{}_{\beta }=\delta ^{\gamma }{}_{\beta }}$	Contracting the indices ${\displaystyle \alpha }$ in 11.
13.	${\displaystyle \Lambda _{\nu }{}^{\gamma }\Lambda ^{\nu }{}_{\beta }=\delta ^{\gamma }{}_{\beta }}$	Contracting the indices ${\displaystyle \mu }$ in 12.
14.	${\displaystyle (\Lambda ^{T})^{\gamma }{}_{\nu }\Lambda ^{\nu }{}_{\beta }=\delta ^{\gamma }{}_{\beta }}$	Swap the order of indices in order to transpose the first ${\displaystyle \Lambda }$ of 13.
15.	∴ ${\displaystyle \Lambda ^{T}=\Lambda ^{-1}}$ ${\displaystyle \Box }$	14 may be paraphrased as ${\displaystyle \Lambda ^{T}\Lambda =I}$.

Reference: http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf, page 9.