Background
I studied Rindler's book on Relativity. Relevant information from this book is available on-line here:
http://www.scholarpedia.org/article/Special_relativity:_mechanics
Given that we work with four-vectors $\mathbf{R} = \left(\vec{r},ct\right)$ where the four-velocity is defined as
$$ \tag{1} \mathbf{U}=\frac{d\mathbf{R}}{d\tau}=\gamma\frac{d\mathbf{R}}{dt}=\gamma\left(u\right)\left(\vec{u},c\right), $$
$\vec{u}$ is the spacial "three-velocity" and the four-acceleration is defined as
$$ \tag{2} \mathbf{A}=\gamma\frac{d\mathbf{U}}{d\tau}=\gamma\frac{d}{dt}\left(\gamma\vec{u},\gamma c\right)=\gamma\left(\dot{\gamma}\vec{u}+\gamma\vec{a},\dot{\gamma}c\right), $$
we can define the four-force as
$$ \tag{3} \mathbf{F}=\frac{d}{ {d\tau } } \mathbf{P}=\frac{d}{ {d\tau}}(m_{0}\mathbf{U})=m_{0}\mathbf{A}+\frac{ {dm}_{0} }{ {d\tau} }\mathbf{U}. $$ The four-force $\mathbf{F}$ is said to be mass-preserving if $dm_0/d\tau$ equals zero, i.e. the force leaves the resting mass of the particle unchanged. Moreover
$$ \tag{4} \mathbf{F}=\frac{d}{d\tau}\mathbf{P}=\gamma(u)\frac{d}{ {dt} }( \vec{p},mc)=\gamma (u)\left(\vec{f},\frac{1}{c}\frac{dE}{dt}\right). $$
We note that $\mathbf{A}\cdot\mathbf{U}=0$ and therefore for a mass preserving four-force $\mathbf{F}\cdot\mathbf{U}=0$ from Eq. (3) or from Eq. (4)
$$ \tag{5} \mathbf{F}\cdot\mathbf{U}=\gamma^2(u)\left(\frac{dE}{dt}-\vec{f}\cdot\vec{u}\right)=0 $$
which means that
$$ \frac{dE}{dt}=\vec{f}\cdot\vec{u} $$
and
$$ \tag{6} \mathbf{F}=\gamma(u)\left(\vec{f},\vec{f}\cdot\vec{u}/c\right). $$
We also have the relationship
$$ \tag{7} \gamma(u)\vec{f}=m_0\frac{d^2\vec{r}}{d\tau^2}. $$
Moreover we have that the proper acceleration $\alpha$ is defined as
$$ \mathbf{A}\cdot\mathbf{A}=\mathbf{A}^2=-\alpha^2. $$
I studied the questions formulated in
Why proper acceleration is $du/dt$ and not $du/d\tau$?
Relativistic factor between coordinate acceleration and proper acceleration
and realized that the spatial vectors can be broken down into a parallel component and an orthogonal component, i.e. $\vec{v}=\vec{v}_\parallel+\vec{v}_\perp$ or $\vec{a}=\vec{a}_\parallel+\vec{a}_\perp$ where one makes a distinction between the component of the velocity that is parallel with the acceleration and vice versa.
Question
Now it can be shown that for a force $\vec{f}$ acting upon a particle traveling with a velocity vector $\vec{u}$ where there is an angle $\theta$ between $\vec{f}$ and $\vec{u}$ the relationship between the proper acceleration of the particle and the force $\vec{f}$ is
$$ \tag{8} \vec{f}^2=m_0^2\frac{\gamma_\parallel^2}{\gamma^2}\alpha^2 $$
where
$$ \tag{9} \gamma_\parallel=\frac{1}{\sqrt{1-u_\parallel^2/c^2}}\;\;\;\text{and}\;\;\;\gamma=\frac{1}{\sqrt{1-u^2/c^2}}. $$
Now how can I derive or prove equation (8)? I have tried every feasible combination of Eq. (1) to (7) without success. I've also tried the more basic variants of the Lorentz transformation
but it gets very confusing when there are two components to consider when assessing the relativistic effects. Any ideas?