Let $(p_1,V_1),(p_2,V_2)$ representation of a lie algebra $\mathfrak g$ on $V_1,V_2$.
I have to prove that:
$ i)\quad$ the direct sum $p_1 \oplus p_2$ is a representation of $\mathfrak g$ in $V_1 \oplus V_2$
$ ii)\;\;\,$ the tensor product $p_1 \otimes p_2$ is a representation of $\mathfrak g$ in $V_1 \otimes V_2$
My idea is (by definition of direct sum):
$$(p_1 \oplus p_2)[x,y](v_1,v_2)=\big(p_1[x,y](v_1) , p_2[x,y](v_1)\big) = \big((p_1(x)p_2(y)-p_1(y)p_1(x))(v_1) , (p_2(x)p_2(y)-p_2(y)p_2(x))(v_2)\big).$$
But i dont know how to continue to prove the bracket identity.
Any help about this, and for $ii)$
For the direct sum we have \begin{align*} (p_1\oplus p_2)([x,y])(v_1,v_2) & = (p_1([x,y](v_1), p_2([x,y](v_2)] \\ & = ([p_1(x),p_1(y)](v_1), [p_2(x),p_2(y)](v_2)] \\ & = [(p_1\oplus p_2)(x), (p_1\oplus p_2)(y)](v_1,v_2). \end{align*}
Given two Lie algebra representations $\rho_i:\mathfrak{g}\rightarrow V_i$, for $i=1,2$, the tensor product $\rho_1\otimes \rho_2$ is the representation given by \begin{align*} \rho_1\otimes \rho_2\colon \mathfrak{g} & \rightarrow \mathfrak{gl}(V_1\otimes V_2), \\ x & \mapsto \Bigl( v_1\otimes v_2 \mapsto (\rho_1(x))(v_1)\otimes v_2+v_1\otimes (\rho_2(x))(v_2)\Bigr). \end{align*} The inner map is first defined on pure tensors $v_1\otimes v_2$, and then is understood to be extended to the whole space by linearity. A similar computation as above shows that $[(\rho_1\otimes \rho_2)(x),(\rho_1\otimes \rho_2)(y)]=(\rho_1\otimes \rho_2)([x,y])$ for all $x,y\in \mathfrak{g}$.