Given an Lie Algebra $L$ (of finite dimension and over an algebraically closed field with zero characteristic) and an ideal $I$, is it truth that 


$rad\left(\dfrac{L}{I}\right)= \pi(rad(L))$,


where $\pi$ is the projection? 
What happens if $L$ isn't finite dimensional? 
And if the field has positive characteristic?



Radical of projection equals projection of radical?
A counter-example for the infinite dimensional case:


Let $V$ be a vector space with a basis $\{e\_i\mid i\in \mathbb{Z}\_+\}$ and $V\_p=\langle e\_i\mid i=1,\dots,p\rangle$. Consider
$$\mathfrak{t}\_\infty:= \{ x \in \mathfrak{gl}(V)\mid x(V\_i)\subset V\_i \} .$$
as a lie subalgebra of $\mathfrak{gl}(V)$.


Let $E\_{ij}$ be the linear transformations on $V$ such that $E\_{ij}(e\_j)=e\_i$ and $E\_{ij}(e\_p)=0$ if $p\neq j$. Then, we have the formula
$$[E\_{ij},E\_{kl}]=\delta\_j^kE\_{il}-\delta\_l^iE\_{kj}.$$


Convince yourself that
$$\mathfrak{h}:=\{x \in t\_\infty \mid x(V) \subset \langle e\_i\mid i>1\rangle \}$$
is an ideal of $\mathfrak{t}\_\infty$.


So, we have that
$$rad( \mathfrak{t}\_\infty / \mathfrak{h})=$$


$$\langle E\_{1j} \mid j>1\rangle / \mathfrak{h}.$$


Suppose, by absurd, that
$$\pi ( rad(\mathfrak{t}\_\infty) ) = $$


$$rad( \mathfrak{t}\_\infty / \mathfrak{h}) =$$


$$\langle E\_{1j}\mid j>1\rangle / \mathfrak{h}.$$
This implies that
$$\langle E\_{1j}\mid j>1\rangle + \mathfrak{h} = $$


$$rad(\mathfrak{t}\_\infty) + \mathfrak{h}.$$


Therefore,
$$\langle E\_{1j}\mid j>1\rangle + \mathfrak{h} \ni E\_{12}=$$


$$x + y \in rad(\mathfrak{t}\_\infty) + \mathfrak{h}$$


for some $x \in rad(\mathfrak{t}\_\infty)$ and $y \in \mathfrak{h}$. Then 


$$rad(\mathfrak{t}\_\infty) \ni $$


$$[E\_{11},x]=$$


$$[E\_{11},E\_{12}]-[E\_{11},y]=$$


$$E\_{12}+0=E\_{12}.$$


This, again by the formula given above, implies that 


$$rad(\mathfrak{t}\_\infty)$$


contains


$$t\_\infty ':=\{x \in \mathfrak{t}\_\infty\mid x(V\_1)=0\textrm{ and } x(V\_i)\subset V\_{i-1}, i>1\}$$
that is a non-solvable ideal of $\mathfrak{t}\_\infty$. A contradiction.


