In Linear Algebraic Groups 2, we defined the Lie algebra of an algebraic group $ G$ to be the Lie algebra of all left-invariant derivations $ D:k[G]\to k[G]$ where $ k[G]$ is the representing algebra of $ G$. However, we were left trying to figure out exactly how a morphism $ \varphi:G\to H$ determines a morphism $ d\varphi:\mathcal{L}(G)\to\mathcal{L}(H)$. It turns out that the answer is slightly tricky, and thus the Lie algebra in terms of left-invariant derivations $ k[G]\to k[G]$ really can’t be the “real” or “most natural” definition!

Now, I could just give the formula for morphisms right away, but I think it would be a bit unmotivated. So before looking at the formula, let us first look at another way of defining the Lie algebra of an algebraic group. In fact, the next defintion is much more natural in that the functoriality of the Lie algebra construction will be clear, whereas in this case it was not.
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