In $SU(N)$, the set of matrices in the fundamental representation plus the identity:
$$ \left\{ \mathbf{1}, t^a \right\} $$
acts as a basis for generator products under matrix multiplication, such that any combination of the elements in the set can be written as a linear combination, i.e. for any
$$ X_F = t^{a_1}\dotsm t^{a_n} $$
we can write
$$ X_F = c^0 \mathbf{1} + c^a t^a \, .$$
It is easy to show that
$$ c^0 = \frac{1}{N} \text{tr} (X_F) \, , \quad c^a = - 2 i\, \text{tr} (X_F t^a)\, . $$
We cannot do the same for the adjoint representation because the set
$$ \left\{ \mathbf{1}, T^a \right\} $$
doesn't span the full product space, e.g. $T^a T^b$ cannot be expressed solely in terms of $\mathbf{1}$ and $T^c$.
So my question is: can we add - and if yes which - a minimal number of $n^2\!-\!1$ matrices $\{K^a\}$ to $ \left\{ \mathbf{1}, T^a \right\} $ in order to make it a closed set?
EDIT
Where it comes from is that because in the fundamental representation we have additional anti commutation rules
$$ \left\{ t^a, t^b \right\} = \delta^{ab} \frac{\mathbf{1}}{N} + d^{abc} t^c \, , $$
we can take the mean of these and the standard commutation rules to write
$$ t^a t^b = \frac{1}{2} \delta^{ab} \frac{\mathbf{1}}{N} +\frac{1}{2} h^{abc} t^c \,,$$
where $h^{abc}=d^{abc}+i f^{abc}$. We chain this relation to write for any product of fundamental generators
$$ t^{a_1}\cdots t^{a_n} = A^{a_1\cdots a_n} \frac{\mathbf{1}}{N} + B^{a_1\cdots a_n b} t^b \, ,$$
with $A$ and $B$ some constants. This is the same statement as above.
However we cannot do the same in the adjoint representation, because then the anti commutation rules don't close, i.e. they cannot be written as a linear combination of $\mathbf{1}$ and $T^a$.
So the question is, can we add some extra matrices $\{K^a\}$ such that we can write the anti commutation rules as
$$ \left\{ T^a, T^b \right\} = \delta^{ab} \frac{\mathbf{1}}{N} + d^{abc} T^c + c^{abc}K^c\, , $$
which would give
$$ T^{a_1}\cdots T^{a_n} = A^{a_1\cdots a_n} \frac{\mathbf{1}}{N} + B^{a_1\cdots a_n b} T^b + C^{a_1\cdots a_n b} K^c\, .$$