I am stuck in deriving a specific formula concerning Lorentz-boosts. In my Classical Mechanics skript there is a chapter dealing with special relativity. In this chapter the Lorentz transformations are defined as being those linear transformations $\Lambda_\nu^\mu:{\mathbb R}^4 \rightarrow \mathbb R^4$, $x^\mu \rightarrow x'^\mu=\Lambda_\nu^\mu x^\nu+\rho^\mu$ that satisfy the equality $g_{\nu\mu}\Lambda_\sigma^\mu\Lambda_\rho^\nu=g_{\sigma\rho}$, where $g_{\nu\mu}$ is the metric tensor for flat space time.
Furthermore a specific transformation between two inertial systems O and O' is considered. This transformation relates the system O' and O if O' moves with velocity $\vec{v}$ relatively to O.
Then, for this specific situation, the formulas $\Lambda_0^0=\gamma$ and $\Lambda_0^i=\gamma \frac{v^i}{c}$ for $i=1,2,3$ are derived. (I appended the derivation in a picture below.)
Now to my question: My script states, that if the axis of both systems are paralell to each other it is also possible to derive formulas for the remaining components of $\Lambda_\nu^\mu$. These formulas are stated to be: $\Lambda_j^0=\gamma \frac{v^j}{c}$ and $\Lambda_j^i=\delta^i_j+\frac{v_iv_j}{\vec{v}^2}\gamma$. The proof for this equations is not carried out in the script but left as an exercis. I have tried really hard but was not able to do the derivation. I consider my problems to be rooted in my inability to figure out how to use the paralellity of the axes in the derivation. I would be really glad if someone could tell me how it is done correctly.
In the following you can find the derivation for $\Lambda_0^i$ and $\Lambda_0^0$ from my skript.