Let $X$ be a smooth projective complex variety and $t\in H^{2p}(X,\mathbf{Q})(p)$ a rational cohomology class.


Assume that there exist
$$t\_1,\ldots,t\_N\in H^{2\*}(X,\mathbf{Q})(\*)$$
algebraic classes (i.e. in the image of the cycle map $cl : A^\*(X)\to H^{2\*}(X,\mathbf{Q})(\*)$) and an algebraic subgroup $$G\subset\text{Aut}(H^\*(X,\mathbf{Q}))\times\text{GL}(\mathbf{Q}(1))$$ (automorphisms as a graded $\mathbf{Q}$-algebra) fixing the $t\_1,\ldots, t\_N$ and largest with respect to this property, such that $G$ also fixes $t$.



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> Is $t$ algebraic?
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> 


From papers of Milne, one can deduce this is true under the variational Hodge conjecture. However, I wonder if this is already known unconditionally.



Cohomology classes fixed by algebraic automorphism subgroups
Essentially you are asking whether the category of homological motives generated by $X$ is tannakian, which is true if and only if homological equivalence coincides with numerical equivalence (Jannsen, Deligne). This coincidence is known for abelian varieties (Lieberman) but not much else, so I think the answer to your question is No in general.


