  We give a stably birational classification for algebraic tori of dimensions
$3$ and $4$ over a field $k$. First, we define the weak stably equivalence of
algebraic tori and show that there exist $13$ (resp. $128$) weak stably
equivalent classes of algebraic tori $T$ of dimension $3$ (resp. $4$) which are
not stably rational by computing some cohomological stably birational
invariants, e.g. the Brauer-Grothendieck group of $X$ where $X$ is a smooth
compactification of $T$, provided by Kunyavskii, Skorobogatov and Tsfasman. We
make a procedure to compute such stably birational invariants effectively and
the computations are done by using the computer algebra system GAP. Second, we
define the $p$-part of the flabby class $[\hat{T}]^{fl}$ as a
$Z_p[Sy_p(G)]$-lattice and prove that they are faithful and indecomposable
$Z_p[Sy_p(G)]$-lattices unless it vanishes for $p=2$ (resp. $p=2,3$) in
dimension $3$ (resp. $4$). The $Z_p$-ranks of them are also given. Third, we
give a necessary and sufficient condition for which two not stably rational
algebraic tori $T$ and $T^\prime$ of dimensions $3$ (resp. $4$) are stably
birationally equivalent in terms of the splitting fields and the weak stably
equivalent classes of $T$ and $T^\prime$. In particular, the splitting fields
of them should coincide if $\hat{T}$ and $\hat{T}^\prime$ are indecomposable.
Forth, for each $7$ cases of not stably but retract rational algebraic tori of
dimension $4$, we find an algebraic torus $T^\prime$ of dimension $4$ which
satisfies that $T\times_k T^\prime$ is stably rational. Finally, we give a
criteria to determine whether two algebraic tori $T$ and $T^\prime$ of general
dimensions are stably birationally equivalent when $T$ (resp. $T^\prime$) is
stably birationally equivalent to some algebraic torus $T^{\prime\prime}$ of
dimension up to $4$.
