Describe all Lorentz transformations in $\R_1^3$.
Let $b_1$ be a unit vector in $e_0^\perp$, i.e. $b_1=\cos(\a)e_1+\sin(\a)e_2$ and $b_2=-\sin(\a)e_1+\cos(\a)e_2$ and assume
$$
u(e_0)=\cosh(\vp)e_0+\sinh(\vp)b_1
$$
Put $v(e_0)=\cosh(\vp)e_0-\sinh(\vp)b_1$, $v(b_1)=-\sinh(\vp)e_0+\cosh(\vp)b_1$ and $v(b_2)=b_2$. Then $u=v^{-1}r$ for some rotation $r$ in $e_0^\perp$. The matrix $V^{-1}$ of $v^{-1}$ with respect to the basis $e_0,b_1,b_2$:
$$
\left(
\begin{array}{ccc}
\cosh(\vp)&\sinh(\vp)&0\\
\sinh(\vp)&\cosh(\vp)&0\\
0&0&1
\end{array}
\right)~.
$$
The matrix $U$ of the basis $e_0,b_1,b_2$ with respect to the basis $e_0,e_1,e_2$:
$$
\left(
\begin{array}{ccc}
1&0&0\\
0&\cos(\a)&-\sin(\a)\\
0&\sin(\a)&\cos(\a)
\end{array}
\right)~.
$$
Finally the matrix $R$ of $r$ with respect to the basis $e_0,b_1,b_2$:
$$
\left(
\begin{array}{ccc}
1&0&0\\
0&\cos(\psi)&-\sin(\psi)\\
0&\sin(\psi)&\cos(\psi)
\end{array}
\right)~.
$$
Thus the matrix of $u=v^{-1}r$ with respect to the basis $e_0,e_1,e_2$ is given by
$$
U^{-1}v^{-1}UU^{-1}RU=U^{-1}v^{-1}RU
$$
i.e.
$$
\left(
\begin{array}{ccc}
1&0&0\\
0&\cos(\a)&\sin(\a)\\
0&-\sin(\a)&\cos(\a)
\end{array}
\right)
\left(
\begin{array}{ccc}
\cosh(\vp)&\sinh(\vp)&0\\
\sinh(\vp)&\cosh(\vp)&0\\
0&0&1
\end{array}
\right)
\left(
\begin{array}{ccc}
1&0&0\\
0&\cos(\psi)&-\sin(\psi)\\
0&\sin(\psi)&\cos(\psi)
\end{array}
\right)
\left(
\begin{array}{ccc}
1&0&0\\
0&\cos(\a)&-\sin(\a)\\
0&\sin(\a)&\cos(\a)
\end{array}
\right)~.
$$
For $\psi=0$ we get the matrix of an arbitrary boost in $\R_1^3$:
$$
\left(
\begin{array}{ccc}
\cosh(\vp)&\cos(\a)\sinh(\vp)&-\sin(\a)\sinh(\vp)\\
\cos(\a)\sinh(\vp)&\sin^2(\a)+\cos^2(\a)\cosh(\vp)&\sin(\a)\cos(\a)(1-\cosh(\vp))\\
-\sin(\a)\sinh(\vp)&\sin(\a)\cos(\a)(1-\cosh(\vp))&\cos^2(\a)+\sin^2(\a)\cosh(\vp)
\end{array}
\right)~.
$$