Are there a set of $3\times3$ involutory matrices that form a basis?

Question

My understanding is that any $2\times2$ matrix can be decomposed into the Pauli matrices and the $2\times2$ identity matrix, which are all involutory. Is there a set of $3\times3$ matrices which are also involutory and form a basis?

Answer 1

$$ \pmatrix{1&0&0\\ 0&1&0\\ 0&0&-1},\ \pmatrix{1&0&0\\ 0&-1&0\\ 0&0&1},\ \pmatrix{-1&0&0\\ 0&1&0\\ 0&0&1}, $$ $$ \pmatrix{0&1&0\\ 1&0&0\\ 0&0&1},\ \pmatrix{0&0&1\\ 0&1&0\\ 1&0&0},\ \pmatrix{1&0&0\\ 0&0&1\\ 0&1&0}, $$ $$ \pmatrix{0&-i&0\\ i&0&0\\ 0&0&1},\ \pmatrix{0&0&-i\\ 0&1&0\\ i&0&0},\ \pmatrix{1&0&0\\ 0&0&-i\\ 0&i&0}. $$