Feel free to gloss `interesting' as you see fit. One way:
1. What is the lowest degree matric Toda bracket in $\pi_*(S)$ that doesn't contain zero?
By `degree', I mean total homotopical degree, i.e. the $*$ in $\pi_*$. By `matric' I mean to exclude ordinary Toda brackets, that is matric Toda brackets all of whose entries are one-by-one matrices, and also to exclude brackets that are trivially determined by ordinary Toda brackets.
I'm also interested in the title question with `first' replaced by `simplest'. For instance:
2. What is the lowest order matric Toda bracket in $\pi_*(S)$, all of whose matrix entries are sums of products of Hopf elements, that doesn't contain zero?
By `order' I mean the number of entries in the bracket, i.e. whether it is 3-fold or 4-fold or 5-fold or ... .
Now I'd like to know the same, but with the proviso that the bracket be detectable in the classical Adams spectral sequence:
3. What is the first or simplest, interesting matric Toda bracket in $\pi_*(S)$ that is detectable in the $\mathrm{E}_2$ term of the classical Adams spectral sequence?
Some remarks:
- I believe the matric Massey product $\langle h_2^2, h_0, \left(\begin{array}{cc} h_1 h_3 & h_2^2 \end{array} \right), \left(\begin{array}{c} h_1 \\ h_2 \end{array}\right)\rangle $ in the $\mathrm{E}_2$ term of the Adams spectral sequence is the class $e_0$, but that is not a permanent cycle.
- Kochman lists the matric Toda bracket $\langle \sigma, \left(\begin{array}{cc} A[31] & \nu\end{array}\right),\left(\begin{array}{cc} \eta & 0 \\ 0 & \eta \end{array}\right),\left(\begin{array}{c} \nu \\ \eta A[30] \end{array}\right)\rangle$ in degree 44 as containing an element of order 8, where $A[30]$ and $A[31]$ refer to certain elements of order 2 in those degrees.
