If $R:=\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ and $S^2:=Spec(R)$ is the real 2-spere, a classical result of Borel and Serre says that the only real spheres with an almost complex structure is $S^2$ and $S^6$. An almost complex structure is an endomorphism of the tangent bundle
$$J: T_{S^2} \rightarrow T_{S^2}$$
with $J^2=-Id$. In the case of the real 2-sphere it follows the tangent bundle $T_{S^2}$ is a real algebraic vector bundle of rank 2.
Question 1: I ask for an explicit formula for the endomorphism $J$ in this case. Is this endomorphism algebraic? I ask for references.
Question 2: I'm also looking for an example of a real algebraic (even dimensional) manifold $M$ with an almost complex structure $J:T_M \rightarrow T_M$, where $J$ is not algebraic. The problem of constructing a holomorphic structure on $S^6$ - is this still an open problem?
Note: If you let $k:=\mathbb{R}$ and $K:=\mathbb{C}$, it follows there is an isomorphism (this was proved in another post on this site):
$$Spec(K\otimes R)\cong Spec(B):=S^2_K$$
with
$$B:=K[u,v,w]/(uv-(w^2+1)).$$
If $J_K$ is the pull-back of $J$ to $S^2_K$ it follows
$$\phi:=\frac{1}{2}(I+iJ) \in End(T_{S^2_K})$$
is an idempotent: $\phi^2=\phi$ and you get a direct sum
$$T_{S^2_K} \cong L_1\oplus L_2$$
with $L_i \in Pic(S^2_K)$. If $J$ is algebraic it follows $L_i$ are algebraic, and Im interested in this decomposition.
In the case of the real 6-sphere $S^6$ it follows the endomorphism bundle $End(T_{S^6})$ is a real algebraic vector bundle of rank $36$. We may consider the subvariety
$$I(S^6):=\{\phi \in End(T_{S^6}): \phi^2=-Id\}$$
and the group scheme $G:=GL(T_{S^6})$. There is a canonical action
$$\sigma: G \times I(S^6) \rightarrow I(S^6)$$
and a "parameter space" $I(S^6)/G$ parametrizing algebraic almost complex structures on $S^6$. Is this construction used in the study of the Hopf problem - the problem of constructing a holomorphic structure on $S^6$? If there is a holomorphic structure on $S^6$ - is this neccessarily algebraic?
$\newcommand{\Numbers}[1]{\mathbf{#1}}$[This was written prior to the edit, and in any case may be more of a comment than an answer. Since it's too long for a comment I'm posting anyway in the hope it's useful to you or to posterity.]
The complex structures on $S^{2}$ and $S^{6}$ come from multiplication on the quaternions and octonions, respectively, and both are real-algebraic. If memory serves, details may be found in Volume 1 of Kobayashi-Nomizu.
Specifically, let $\Numbers{H}$ be the algebra of quaternions equipped with the Euclidean structure, $\Numbers{H}_{0}$ the subspace of pure imaginary quaternions, and $S^{2} \subset \Numbers{H}_{0}$ the unit sphere. Each point of $S^{2}$ has the form $p = ai + bj + ck$ for some real numbers $a$, $b$, $c$ with $a^{2} + b^{2} + c^{2} = 1$. The complex structure $J_{p}:T_{p}S^{2} \to T_{p}S^{2}$ is defined by $J_{p}(v) = p \times v$. You can think of $\Numbers{H}_{0}$ as Euclidean three-space, and $J_{p}$ as "taking the cross product with $p$" as an endomorphism of $T_{p}S^{2}$. Familiar properties of the cross product guarantee $J_{p}^{2} = -I_{p}$ on $T_{p}S^{2}$. Stereographic projection from the north pole, together with stereographic projection from the south pole followed by reflection (complex conjugation), show that $J$ comes from a holomorphic structure on $S^{2}$.
Analogously, let $\Numbers{O}$ be the algebra of octonions equipped with the Euclidean structure, $\Numbers{O}_{0}$ the subspace of pure imaginary octonions, and $S^{6} \subset \Numbers{O}_{0}$ the unit sphere. The complex structure $J_{p}:T_{p}S^{6} \to T_{p}S^{6}$ is defined by octonion multiplication $J_{p}(v) = p \times v$. As is well-known, this complex structure does not define a holomorphic structure on $S^{6}$.
(Note: I'm using complex instead of "almost-complex" and holomorphic instead of "complex".)