The following is on page 3 of Introduction to Lie Algebras and Representation Theory by Humphreys:
Here the author claims that the dimension of the orthogonal algebra is $2l^2+l$; but I think the argument is valid only if the characteristic of the field $F$ is not $2$. When $\mathrm{char}(F)=2$, there is no restriction on $a$ and the diagonal elements of $m$ and $q$ and therefore, the dimension should be $(2l^2+l)+(1+2l)$. Am I right?
Yes, you are right. On the other hand, note that the first words of the that textbook are “This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic $0$”. So, there is no reason for the author to mention what occurs in characteristic $2$.