If we have $R^{4}$ with basis $\{e\_{1},e\_{2},e\_{3}=Je\_{1},e\_{4}=Je\_{2}\}$, then we know that $\{e\_{1},e\_{2}\}$ is totally real minimal submanifold of $R^{4}$. Is there a nontrivial example of totally real minimal submanifold of $R^{4}$.


Also, the above totally real minimal immersion is an inclusion map. How we can write a basis of a submanifold in the case when the immersion $f:R^{2}\to R^{4}$ is defined as $f:(x,y)\to (x,y,x,y)$.



Examples of totally real minimal submanifolds in complex space forms
Yes, there are many such examples. 


A totally real submanifold in this case is simply a surface $S\subset\mathbb{R}^4$ such that, for each $p\in S$, the space $J(T\_pS)$ is orthogonal to $T\_pS$. In other words, if $x^i$ are the coordinates dual to your basis $e\_i$, then the $2$-form $\Upsilon\_1 = dx^1\wedge dx^3 + dx^2\wedge dx^4$ should vanish on $S$. Suppose that we also require that the $2$-form $\Upsilon\_2 = dx^1\wedge dx^4 - dx^2\wedge dx^3$ should vanish on $S$. These are equivalent to the vanishing of 
$$
\Upsilon\_1 + i\,\Upsilon\_2 = (dx^1 - i\,dx^2)\wedge (dx^3 + i\,dx^4) = dz^1\wedge dz^2
$$
where $z^1 = x^1-i x^2$ and $z^2 = x^3+ix^4$ are complex coordinates with respect to a different complex structure $J'$ on $\mathbb{R}^4$ than the one you defined, but that is also orthogonal with respect to the underlying metric on $\mathbb{R}^4$. 


Any $J'$-complex curve (and there are many of them, just take $z^2 = h(z^1)$ where $h$ is holomorphic, for example), is then an example of a surface on which $\Upsilon$ vanishes. All $J'$-complex curves are minimal (since $J'$ is orthogonal), so this furnishes many examples of nontrivial totally real minimal surfaces in $\mathbb{R}^4$.


