  One approach to make progress on the symbolic determinant identity testing
(SDIT) problem is to study the structure of singular matrix spaces. After
settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson,
Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a
natural next step is to understand singular matrix spaces whose non-commutative
rank is full. At present, examples of such matrix spaces are mostly sporadic,
so it is desirable to discover them in a more systematic way.
  In this paper, we make a step towards this direction, by studying the family
of matrix spaces that are closed under the commutator operation, that is matrix
Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the
complex number field give rise to singular matrix spaces with full
non-commutative ranks. On the other hand, we show that SDIT of such spaces can
be decided in deterministic polynomial time. Moreover, we give a
characterization for the matrix Lie algebras to yield a matrix space possessing
singularity certificates as studied by Lov'asz (B. Braz. Math. Soc., 1989) and
Raz and Wigderson (Building Bridges II, 2019).
