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Take units $c=1$. You have $U_0^2-\vec U^2=1$, that is $\gamma^2(1-\beta^2)=1$. With some basic transformations, you will get : $\frac{\gamma - 1}{\beta^2}= \frac{\gamma^2}{\gamma + 1}$ Now, from your Wikipedia matrix, you have obvious term, $U_0 =\gamma , U_i =\gamma \beta_i$ You have \$(\gamma -1) \frac{\beta_i\beta_j}{\beta^2} = \frac{\gamma^2}{\gamma ...