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?



Cayley Transform for all reductive groups a.k.a an algebraic logarithm
Bardsley and Richardson (Etale slices for algebraic transformation groups in characteristic p.
Proc. London Math. Soc. (3) 51 (1985), no. 2, 295–317) give a construction which seems to do what you want. It gives less than a "Cayley map" as in Borovoi's answer.


Let $G$ be connected and semisimple over a field of char. 0.
Bardsley and Richardson construct a mapping $G \to \operatorname{Lie}(G)$ with 
nice properties (which I'll indicate below).


Note that we may as well suppose that $G$ is of adjoint type -- indeed,
if the problem is solved already for the adjoint group $G\_{\operatorname{ad}}$,
just take the composite
$$G \to G\_{\operatorname{ad}} \to \operatorname{Lie}(G).$$


Now, since the characteristic of $k$ is zero and $G$ is of adjoint type, the adjoint representation
$V = \operatorname{Lie}(G)$ is a faithful representation of $G$ for which the
trace form
defined by $\kappa(X,Y) = \operatorname{tr}(X \circ Y)$ -- a non-degenerate
form on $\mathfrak{gl}(V)$-- remains non-degenerate on the image
$\operatorname{ad}(\operatorname{Lie}(G)) \simeq \operatorname{Lie}(G) \subset \mathfrak{gl}(V)$.


Writing $M$ for the orthogonal complement $M=\operatorname{ad}(\operatorname{Lie}(G))^\perp$ with respect to the form $\kappa$,
we have
$$\mathfrak{gl}(V) = M \oplus \operatorname{ad}(\operatorname{Lie}(G))$$
as $G$-representations.
Write $\pi:\mathfrak{gl}(V) \to \operatorname{ad}(\operatorname{Lie}(G))$ for
the projection on the second factor.


Since $G$ is semisimple, $\operatorname{ad}(\operatorname{Lie}(G)) \subset
\mathfrak{sl}(V)$ so that the identity mapping $I$ satisfies
$\kappa(I,\operatorname{ad}X) = \operatorname{tr}(\operatorname{ad}X) = 0$ 
for each $X \in \operatorname{Lie}(G)$. Thus $I \in M$.


Write $\lambda$ for the composite mapping
$$G \to \operatorname{GL}(V) \subset \mathfrak{gl}(V) \xrightarrow{\pi} \operatorname{ad}(
\operatorname{Lie}(G))$$.


Since $I \in M$, evidentally $\lambda(1) = 0$. Since $\pi$ is a $G$-module
homomorphism, $\lambda$ is $G$-equivariant.
Moreover, by construction $d\lambda\_1$ is the identity mapping. Finally, by $G$-equivariance
the image under $\lambda$ of a maximal torus $T$ is contained in the $T$-fixed
points of $\operatorname{Lie}(G)$, i.e. in $\operatorname{Lie}(T)$.


This verifies the stipulated conditions (1),(2),(3) and (4).


Note that Barsdsley and Richardson go on to show that the restriction of $\lambda$
to the unipotent variety $\mathcal{U} \subset G$ defines a $G$-equivariant isomorphism
$\mathcal{U} \xrightarrow{\sim} \mathcal{N}$ where $\mathcal{N} \subset \operatorname{Lie}(G)$ is the nilpotent variety.


Moreover, by Luna's theorem (a proof valid in positive characteristic is given in Bardsley and Richardson's paper) there are $G$-invariant open subset $U \subset G$ and $U' \subset \operatorname{Lie}(G)$ with $1 \in U$ and $0 \in U'$ such that $\lambda\_{\mid U}$ defines a surjective
etale mapping $U \to U'$. So $\lambda$ need not be birational, but it is fairly nice.


Note that Barsdley and Richardson actually formulate the above construction more generally
using representations $V$ of $G$ which are "nice enough" (among other things, the restriction of the traceform on $\mathfrak{gl}(V)$ to the image of $\operatorname{Lie}(G)$ must be non-degenerate), and under suitable assumptions (very roughly: the characteristic should be good for $G$) their construction gives "explicit" Springer isomorphisms $\mathcal{U} \xrightarrow{\sim} \mathcal{N}$ in characteristic $p>0$.


