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Is it true that for every reductive algebraic $G$ over ${\mathbb C}$ with a Lie algebra $\mathfrak g$ there is an open neighborhood $U$ of the identity in $G$ and an algebraic function (in a sense of algebraic geometry) $L: U\to \mathfrak g$ which satisfies the following properties of logarithm:

(1) $L$ is $G$-equivariant with respect to the $G$-action on $G$ by conjugation and the Adjoint $G$-action on $\mathfrak g,$
(2) $L(e)=0,$
(3) $dL$ is an isomorphism at $e$,
(4) For some maximal torus $T$ in $G$, $L(T\cap U)$ lies in the Lie algebra of $T.$

For $G=GL(n,\mathbb C)$, the embedding $L:GL(n,\mathbb C)\to gl(n,\mathbb C)$ works.
For $G=SO(n,\mathbb C)$, the Cayley Transform works: $L(A)= (I-A)(I+A)^{-1}$.
Cayley transform has a version for symplectic matrices as well.

Is there a construction which works for all $G$? If not, are there known ad hoc constructions for exceptional groups?

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# Cayley Transform for all reductive groups a.k.a an algebraic logarithm

Is it true that for every reductive algebraic $G$ over ${\mathbb C}$ with a Lie algebra $\mathfrak g$ there is an open neighborhood $U$ of the identity in $G$ and an algebraic function (in a sense of algebraic geometry) $L: U\to \mathfrak g$ which satisfies the following properties of logarithm:

(1) $L$ is $G$-equivariant with respect to the $G$-action on $G$ by conjugation and the Adjoint $G$-action on $\mathfrak g,$
(2) $L(e)=0,$
(3) For some maximal torus $T$ in $G$, $L(T\cap U)$ lies in the Lie algebra of $T.$

For $G=GL(n,\mathbb C)$, the embedding $L:GL(n,\mathbb C)\to gl(n,\mathbb C)$ works.
For $G=SO(n,\mathbb C)$, the Cayley Transform works: $L(A)= (I-A)(I+A)^{-1}$.
Cayley transform has a version for symplectic matrices as well.

Is there a construction which works for all $G$? If not, are there known ad hoc constructions for exceptional groups?
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