Lie algebras of reductive groups Let $k$ be an algebraically closed field of positive characteristic and let $G$ be a connected split reductive group.  We know $G$ is the product of its center $Z(G)$ and derived group $[G, G]$ and $[G, G]$ is semisimple so it's a product of simple algebraic groups $G_1, \ldots, G_n$ corresponding to the decomposition of it's root space.  So $G = G_1\cdots G_nZ(G)$.
My questions are about the Lie algebras of these groups.


*

*Is it true that $\mathrm{Lie}([G, G]) = \mathrm{Lie}(G_1) \times \cdots \times \mathrm{Lie}(G_n)$?

*Is it true that $\mathrm{Lie}(G) = \mathrm{Lie}([G, G]) \times \mathrm{Lie}(Z(G))$?
I think the first is true.  The dimension is right and the $G_i$ commute with each other so their Lie algebras should commute as well.  I'd like the second to be true as well for basically the same reason.
I'm a little worried that maybe something funny can happen due to the products not being direct products (there can be a finite intersection which if taken scheme theoretically might have a Lie algebra?) and also maybe there should be a condition on the characteristic of the field.  All the references that discuss these structure theorems (I have Malle and Testerman and all the LAG's: Humphreys, Borel, Springer) never talk about what this means for the Lie algebras.  I'd be happy with a straight up answer or a reference if anyone knows of one.
 A: Edit: There are some issues with the ideas in this answer discussed in the comments.
As was already discussed in the comments, 1. is clear if $[G,G]$ is a direct product of simple algebraic groups, in particular, if $[G,G]$ is simply connected.
Note that for any semisimple group $G$, we have an isogeny $\pi : G_{sc} \rightarrow G$ inducing an isomorphism on the Lie algebras (this is Proposition 9.15 in Malle-Testerman), so $\text{Lie}(G) \cong \text{Lie}(G_{sc})$.
We have $[G,G] = G_1\cdots G_n$ a product decomposition into simple algebraic groups $G_i$ and consequently $[G,G]_{sc} = (G_1)_{sc} \times \cdots \times (G_n)_{sc}$. Using the isomorphisms between Lie algebras induced by $[G,G]_{sc} \rightarrow [G,G]$ and $(G_i)_{sc} \rightarrow G_i$ we get $$ \text{Lie}([G,G]) \cong \text{Lie}([G,G]_{sc}) \cong \prod_{i = 1}^n \text{Lie}((G_i)_{sc}) \cong \prod_{i = 1}^n \text{Lie}(G_i)$$ and this gives 1.
For any connected reductive $G$ we have a surjective morphism $[G,G] \times Z(G) \rightarrow G$ of algebraic groups given by multiplication. The kernel of this morphism is finite, hence its Lie algebra is trivial (e.g. Thm. 7.4 a in Malle-Testerman) giving an isomorphism of Lie algebras (by Thm. 7.9 in Malle-Testerman)  $$\text{Lie}(G) \cong  \text{Lie}([G,G] \times Z(G)) \cong \text{Lie}([G,G]) \times \text{Lie}(Z(G)).$$
