I've recently met with the [Temperley-Lieb algebra](https://en.wikipedia.org/wiki/Temperley%E2%80%93Lieb_algebra) in my work. I'm in no way a specialist, and it's seems like a pretty simple question, but nevertheless. I'm interested in the subalgebra generated by two neighbour generator $ U\_i, \, U\_{i+1}$. This subalgebra is 5 dimensional with a basis:
$$
\mathbb{1}, \, U\_i, \, U\_{i+1}, \, U\_i U\_{i+1}, \, U\_{i+1} U\_i.
$$
It seems very simple, and it'll be very helpful to me if the subalgebra is isomorphic to a well-known matrix algebra, for example. I've looked throught references, but mostly they talk about representation-theoretical questions, and simple algebraic ones like mine are not discussed.


Therefore, my question is the following. Is the Temperley-Lieb subalgebra generated by two neighbor generators isomorphic to some well-known algebra? Or is there any way I can check it myself?



Subalgebras of the Temperley-Lieb algebra
As YCor mentions in the comments, this is (isomorphic to) the algebra $TL\_3$, so I will call its generators $U\_1,U\_2$. I will henceforth write $R := TL\_3$. I do not know what your ground ring is or what your parameter $\delta$ is. If $1-\delta^2$ is invertible, then $R$ contains a special idempotent $p$ named after Jones and Wenzl. The Jones–Wenzl idempotent is very special: $U\_1 p = p U\_1 = U\_2 p = p U\_2 = 0$. In particular it is central. It follows that your ring splits as $pR \times (1-p)R$. A formula is
$$ p = 1 - \frac{\delta}{\delta^2 - 1} ( U\_1 + U\_2) + \frac{1}{\delta^2-1} (U\_1 U\_2 + U\_2 U\_1). $$


Since $U\_ip = 0$, $pR$ is definitely 1-dimensional, and so $R' := (1-p)R$ is a noncommutative 4-dimensional ring. I claim it is a matrix ring. Note that $R'$ is spanned for example by the elements $1-p$, $U\_1$, $U\_2$, and $U\_1U\_2$. Consider the left $R'$-ideal $R'U\_1 \subset R'$. It is 2-dimensional: the basis vectors map to $U\_1, \delta U\_1, U\_2 U\_1$, and $U\_1$ respectively, and so we can take the basis $\{U\_1, U\_2 U\_1\}$ of this ideal. In this basis, the action of $R'$ on the ideal is
$$ 1-p \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad U\_1 \mapsto \begin{pmatrix} \delta & 1 \\ 0 & 0 \end{pmatrix}, \quad U\_2 \mapsto \begin{pmatrix} 0 & 0 \\ 1 & \delta \end{pmatrix}, \quad U\_1 U\_2 \mapsto \begin{pmatrix} 1 & \delta \\ 0 & 0 \end{pmatrix}. $$
Since I insisted that $1-\delta^2$ should be invertible, these four matrices are a basis for the $2\times 2$ matrix ring, proving the claim.


